order-theoretic metrical fixed point theorems with

ORDER-THEORETIC METRICAL FIXED POINT THEOREMS WITH APPLICATIONS THESIS SUBMITTED FOR THE AWARD OF THE DEGREE OF

DOCTOR OF PHILOSOPHY IN

MATHEMATICS

BY

ABDULRQEEB MOHSEN AHMED (Rqeeb Gubran)

U NDER THE S UPERVISION OF

PROF. MOHAMMAD IMDAD

DEPARTMENT OF MATHEMATICS

ALIGARH MUSLIM UNIVERSITY ALIGARH - 202002, INDIA S EPTEMBER -2018

c Aligarh Muslim University, Aligarh, 2018.

Dedicated to Memories of my grandparents. My parents, who have been a constant source of support and encouragement during the challenges of my life. My dearest wife and my children Mohsen, Eman, Ahmed, Haneen and Reem, whom I am truly thankful for having you in my life. My beloved brothers and sisters. All the people in my life who touch my heart.

Contents

Acknowledgements

v

List of Abbreviations & Symbols

vii

List of Publications

ix

Preface

x

1 Introduction and Preliminaries

1

1.1

An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Metric fixed point theory . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.1

Certain Generalized Linear Contractions . . . . . . . . . . . .

5

1.2.2

Nonlinear Contractions . . . . . . . . . . . . . . . . . . . . . .

7

1.2.3

Weak Contractions: . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2.4

Unified Fixed points Results . . . . . . . . . . . . . . . . . . . .

9

1.2.5

Theory of Coincidence Points . . . . . . . . . . . . . . . . . . . 12

1.3

Order-Theoretic fixed point theory . . . . . . . . . . . . . . . . . . . . 14 1.3.1

Classical Order-Theoretic Notions . . . . . . . . . . . . . . . . 15

1.3.2

Order-Theoretic Fixed Point Results . . . . . . . . . . . . . . . 19

2 Results for Boyd-Wong Type Contractions

23

2.1

Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . 23

2.2

Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3

Uniqueness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4

An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Results for Matkowski Type Contractions

iii

33

3.1


3.2

Fixed point Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3

3.2.1

Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2

Uniqueness Results . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.3

An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Common Fixed point Results . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.1

Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2

Uniqueness Results . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Results for Generalized Weak Contractions

55

4.1


4.2

Coincidence Point Results . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3

Common Fixed Points Results . . . . . . . . . . . . . . . . . . . . . . . 65

4.4

An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Results Under an Implicit Function

69

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2

Implicit function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3

Common Fixed Point Results . . . . . . . . . . . . . . . . . . . . . . . 74

5.4

Corresponding Results on Metric Spaces . . . . . . . . . . . . . . . . . 86

5.5

An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 Results Under F Function

89

6.1


6.2

An Observation on α-Type F-Contractions . . . . . . . . . . . . . . . . 93

6.3

Fixed Point Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4

An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7 Results Under Simulation Function

107

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2

Simulation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3

Common Fixed Point Results . . . . . . . . . . . . . . . . . . . . . . . 111

7.4

An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Bibliography

121 iv

Acknowledgments First and foremost, I must bow in reverence to Almighty ’ALLAH’ the cherisher and the sustainer to whom we entirely owe our all capabilities, strength, courage, knowledge and belive that without his blessing nothing can be achieved.

I would like to express my sincere gratitude to my thesis supervisor, Prof. Mohammad Imdad, for his guidance and encouragement during the period of my research. His dynamism, vision, sincerity and motivation have deeply inspired me. I am grateful to him for having a lot of impact on my professional and personal developments. This thesis would not be in this form without his expert guidance and kind inspirations.

I would like to thank Prof.

M. Mursaleen, Chairperson, Department of

Mathematics, AMU Aligarh, for providing necessary facilities for carrying out the research and for his help whenever I approached him. I would like to thank every member of the Department for creating a healthy and cooperative environment to carry out the research activities.

I must express my very profound gratitude to Prof. Imdad’s group, Dr. QH Khan, Dr. J. Ali, Dr. Izhar Uddin, Dr. A. Alam, Mr. Md Ahmadullah, Mr. M. Arif, Mr. W. M. Alfaqih, Mr. M. Asim, Ms. A. Parveen, Mr. Hayel and Mr. Hassanuzzaman for their help. Also, special thanks to all my friends, especially Mr. Hamzah Gamel for his help and constant encouragements.

I would like to thank my parents, my brothers and sisters for supporting me spiritually throughout writing this thesis and for all their love. I am deeply indebted to my wife and children as I was not able to take care of them properly during the period of my research. I am very much thankful to them for their love, understanding, prayers and constant support to complete this work. v

Finally, I would like to thank all those people who helped me directly or indirectly in completing my research work at AMU Aligarh.

Abdulrqeeb Mohsen Ahmed

( RqeebGubran)

vi

List of Abbreviations & Symbols et al.

et alii (and others).

etc.

et cetera "and other things" or "and so on".

e.g.

exempli gratia (for example).

i.e.

id est (that is).

resp.

respectively.

w.r.t.

with respect to.

6=

non-equality.

≡

equivalent.

∅

empty set.

∈, 6 ∈

belongs to and does not belong to.

⇒

implication.

⇔

if and only if.

max

maximum.

min

minimum.

inf

infimum (or greatest lower bound).

sup

supremum (or least upper bound).

N for all n

the set of natural numbers. for all n ∈ N.

N0

set of whole numbers.

Q

set of rational numbers.

R

set of real numbers.

Rn

n-dimensional Euclidean space.

|u|

absolute value of real number u. vii

lim ϕ(r )

right hand limit of the function ϕ.

lim inf ϕ(r )

right hand lower of the function ϕ.

lim sup ϕ(r )

right hand upper limit of the function ϕ.

r →t+ r →t+

r →t+

lim un , lim sup un

n→∞

n→∞

−→ d

−→

limit and upper limit of the sequence {un }. convergence in real line. convergence in metric space ( M, d).

M

nonempty set.

IM

the identity mapping on the set M.

S or T S(u) or Su

mappings (usually on the set M). image of u under S.

S( M)

range of S.

Fix (S)

set of fixed points of S.

Sn

So f n−1 .

C(S, T )

the set of all coincidence points of S and T.

C(S, T )

the set of all points of coincidence of S and T. n o d( Tu,Sv)+d( Tv,Su) max d( Tu, Tv), d( Tu, Su), d( Tv, Sv), 2 o n d(u,Sv)+d(v,Su) max d(u, v), d(u, Su), d(v, Sv), 2 n o d( Tu,Su)+d( Tv,Sv) d( Tu,Sv)+d( Tv,Su) max d( Tu, Tv), , 2 2 n o d(u,Su)+d(v,Sv) d(u,Sv)+d(v,Su) max d(u, v), , 2 2

MS,T (u, v) MS (u, v) mS,T (u, v) mS (u, v)

, ≺

precedes and strictly precedes.

,

succeeds and strictly succeeds.

≺

comparability.

un ↑ u

{un } is increasing and converging to u.

un ↓ u

{un } is decreasing and converging to u.

un ↑↓ u

{un } is monotone and converging to u.

un l u

{un } is term-wise monotone and converging to u

viii

List of Publications The content of this thesis is based on the following articles 1. M. Imdad and R. Gubran Ordered-theoretic fixed point results for monotone generalized Boyd-Wong and Matkowski type contractions Journal of Advanced Mathematical Studies 10 (1) (2017), 49-61. 2. M. Imdad and R. Gubran and Md, Ahmadullah Using an implicit function to prove common fixed point theorems Journal of Advanced Mathematical Studies 11 (3) 2018, 481–495. 3. M. Imdad, R. Gubran, M. Arif and D. Gopal An observation on α-type Fcontractions and some ordered-theoretic fixed point results Mathematical Sciences 11 (3) 2017, 247–255. 4. R. Gubran, W. M. Alfaqih and M. Imdad Common fixed point results for αadmissible mappings via simulation function The Journal of Analysis, 25 (2) 2017, 281–290. 5. R. Gubran and M. Imdad Results on Coincidence and Common Fixed Points for (ψ, ϕ) g -Generalized Weakly Contractive Mappings in Ordered Metric Spaces Mathematics, 4 (68) 2016. 6. R. Gubran, I. A. Khan and M. Imdad Fixed point theorems for Matkowski-type nonlinear contractions in ordered metric spaces Journal of Inequalities and Special Functions 9 (3) 2018.

ix

Conferences 1. International Conference on on Algebra and its Applications ICAA-14. AMU. India 15–17 Dec. 2014. (participation) 2. International Conference on on Algebra and its Applications ICAA-16. AMU. India 12–14 Nov. 2016. (participation) 3. International Conference on on Analysis and its Applications ICAA-17. AMU. India 20–22 Nov. 2017. (Presentation) 4. International Conference on Mathematical Analysis and its Applications ICMAA-2017 Lature, India 5–9 Mar. 2017. (presentation) Training & Workshops 1. Annual Foundation Schools (AFS-I), Organized by IIT-Bombay at (IISERThirupampuram) 5–31 Dec. 2016. 2. Fixed Point Theory in Probabilistic and Fuzzy Structures, Guru Ghasidas Vishwavidyalaya, Bilaspur, India. 23–27 Oct. 2017. 3. International Workshop on Convex Analysis and Optimization, IWCAO-2017. (AMU, India) 14–19 Nov. 2017. 4. Monotone Iterative Techniques, IIT Patna, India. 5–11 Nov. 2017.

x

Preface Nonlinear analysis is a major branch of mathematics wherein fixed point theory lies in its heart. Indeed, fixed point theory offers an elementary, vigorous and effective tool for nonlinear analysis. It has fruitful applications in mathematics as well as various disciplines of sciences, such as Physics, Chemistry, Computer and several others. Consequently, this theory has gained a remarkable scope of research and attracted many researchers leading the development of this theory in several directions.

Historically, the roots of fixed point theory can be traced back in the mathematical activities of great mathematicians namely: Cauchy [46], Liouville [111], Lipschitz [112], Peano [137], Picard [138], Poincaré [141] and some others. In fact, the initiation of fixed theory can be traced back formally to the beginning of the twentieth century in the pioneer article of the Dutch mathematician Brouwer [43]. However, the proofs in all aforementioned articles were non-constructive.

Metric fixed point theory is a subbranch of fixed point theory containing methods and results that involve properties of an essentially isometric nature. The first result of this setting was in 1922 by the Polish mathematician Banach [31] which, often, refereed as Banach contraction principle. This principle ensures the existence of a unique fixed point for every contraction mapping defined on a complete metric space. Though, Banach principle is very simple and natural, yet the involved condition of contraction is very restrictive.

xi

Thus far, a multitude of results enriching this principle have been proved and such interest is a still on. One of the generalizations of this principle can be obtained by considering a relatively more general contraction condition. For this kind of work, one may recall Chatterjea [48], C’irić [53], Hardy and Rogers [76], Kannan [98, 99], Reich [151] and Sehgal [165]. Here, it can be pointed out that Boyd and Wong [41], Browder [44], Jotic [93], Matkowski [115] and Mukherjea [121] obtained more general contraction conditions using different types of control functions.

Improving Banach contraction principle received a new impetus when the researchers attempted to prove unified-fixed point results. With such a quest, Popa [142] initiated the idea of implicit function. Another effort of this kind is essentially due to Wardowski [187] wherein the idea of F-contraction was initiated. Khojasteh et al. [105] introduced the notion of simulation function which was also designed to unify several contractions.

A notable idea of improving Banach contraction principle is due to Samet et al. [159] wherein authors introduced the idea of admissible mappings which is good enough to extend, generalize and improve many existing results and also yielding order-theoretic results.

Another way of generalizing Banach contraction principle is to prove coincidence or common fixed point results which can be done by increasing the number of involved mappings. The first result concerning coincidence point was by Machuca [114] in 1967, which was further improved by Goebel [70] while Jungck [94] proved the first ever common fixed point theorem in 1976.

Another way of generalizing this principle can be obtained by considering relatively larger classes of (defining) spaces, e.g., b-metric space [30, 60], cone metric space [79], partial metric space [117] and several others. One such important class is that of an ordered metric space. Although such spaces are in fashion only recently, yet xii

these spaces were introduced and studied earlier, e.g., Knaster [107], Tarski [175], Vandergraft [181] and Wolk [189]. With the similar motivation, in 2004, Ran and Reurings [150] obtained a variant of Banach contraction principle for continuous monotone mappings in ordered metric spaces and also presented some applications to a system of non-linear matrix equations. Afterward, Nieto and Rodríguez-López [127, 128] slightly improved Ran-Reurings’ fixed point theorem for monotone mappings (not necessarily continuous) besides presenting some applications to ordinary differential equations.

In this dissertation, motivated by all above-mentioned types of generalizations of Banach contraction principle, we prove some fixed (coincidence) point theorems in the setting of ordered metric spaces and also explore the possibility of their applications. Each chapter of the thesis is divided into various sections and consistently the first section of each chapter provides an introduction to the content of the chapter.

The corollaries, definitions, examples, lemmas, propositions, remarks, theorems etc. have been individually specified with a triple-decimal number. The first figure denotes the number of the chapter while the second figure refers the section and the third represents the number of the items . For example, Theorem 1.2.3 indicates Theorem 3 which can be found in Section 2 of the first chapter. As usual, the numbers in square brackets refer to the references listed in the bibliography.

The contents and characterizations of each chapter can be, briefly, described as follows:

In the first chapter, as usual, we collect the necessary background materials which includes basic definitions, preliminary notions and relevant core results needed in the subsequent chapters.

We also include very briefly historical

development of Metric fixed point theory. Particularly, we focus on ordered metric xiii

spaces and related notions to make our presentation as self-contained as possible.

In Chapter 2, we prove some order-theoretic fixed point theorems for monotone generalized contractions wherein the utilized control function is of Boyd-Wong type. The newly obtained results generalize several known results of the existing ´ c et al. [58], Nieto and literature especially those contained in Agarwal et al. [8], Ciri´ Rodríguez-López [127, 128], Ran and Reurings [150] and Wu and Liu [191]. We furnish several examples to illustrate the usability of our results especially one of these examples disproves one of the remarks contained in Agarwal et al. [8]. As an application, we prove a fixed point result for a mapping satisfying Boyd-Wong integral type contraction in ordered metric space.

The aim of Chapter 3 is of two-fold: firstly, to prove some existence and uniqueness order-theoretic fixed point theorems for monotone generalized Matkowski contractions. As an application of our newly obtained results, we propose a result for a mapping satisfying Matkowski integral type contraction condition. Secondly, to prove some order-theoretic common fixed points results for T-increasing Matkowski contractions.

Inspired by a metrical-fixed point theorem due to Choudhury et al. [50], in Chapter 4, we prove some order-theoretic common fixed point results which generalize several core results of the existing literature, especially the two main results of Harjani and Sadarangani [77, 78]. We demonstrate the realized improvements in our results using an illustrative example. As an application of our main result herein, we prove a result for a pair of mappings satisfying an integral type (ψ, ϕ) T -generalized weak contractive condition.

In Chapter 5, we prove order-theoretic coincidence and common fixed point theorems for a pair of self-mappings satisfying a unified-type condition governed by an implicit function which is general enough to cover a multitude of known as xiv

well as unknown contractions. Our results modify, unify, extend and generalize many relevant results of the existing literature. Interestingly, unlike several other cases, our main results deduce a nonlinear order-theoretic version of a well-known ´ c [55] proved for quasi-contraction. Moreover, we fixed point theorem due to Ciri´ observe that Theorem 2 contained in Berinde and Vetro [38] is not correct in its present form. Finally, in the setting of metric spaces, we drive a sharpened version of Theorem 1 contained in [38]. As an application, we establish an existence result for a solution of Volterra type integral equation.

We observe that some results involving α-type F- contractions of the existing literature are not correct in their present forms. Starting from this point, in Chapter 6, we prove some order-theoretic fixed point results for extended F-weak contraction. Our observations and the usability of our results are substantiated using suitable examples. As an application, we prove an existence and uniqueness result for the solution of a first-order ordinary differential equation satisfying periodic boundary conditions in the presence of either its lower or upper solution.

In 2015, Khojasteh et al. [105] introduced the class of simulation functions and utilized the same to unify several fixed point results of the existing literature. Later on, Karapınar [100] enlarged this class to cover α-admissible contractions. Motivated by aforementioned articles, in Chapter 7, we establish common fixed point results for α-admissible mappings satisfying a nonlinear contraction condition under simulation function. As an application, we establish the existence of solution for certain two-point boundary value problem of second order differential equation.

At the end, a bibliography is given which by no means is exhaustive one but lists only those books and papers which have been referred to in this thesis. Notice that the references in the bibliography are arranged alphabetically according to the last names of the authors.

xv

CHAPTER

Introduction and Preliminaries 1.1

1

An Overview

A self-mapping S on a non-empty set M is said to have a fixed point u ∈ M if u remains invariant under S. If such u does not exist, S is said to be a fixed point free. The branch of mathematics which addresses the set of conditions on M as well as S to ensure the existence of fixed points (and concerns properties of these fixed points) is termed as fixed point theory. Although a mapping may have one, two or even infinitely many fixed points, mathematicians are interested in mappings that have a unique fixed point, that is, when the set of fixed points of S is a singleton set. Fixed point theory remains a very powerful and popular tool in modern mathematics especially in existence and uniqueness considerations. Indeed, solving a system of equations, in general, can be reduced to the problem of determining the fixed points of a self-mapping. To clarify the situation, consider the equation u2 − 3u + 2 = 0. It is easy to see the equivalence between solving this equation and finding u such that u2 + 2 = u. 3 Define a mapping S by Su =

u2 +2 3 ,

(1.1)

then equation (1.1) can be written as Su = u. As

u = 1 and u = 2 are fixed points of S, the problem of finding the solutions of the functional equation Su − u = 0 is the same as finding the fixed points of S. 1

2

C HAPTER 1: I NTRODUCTION AND P RELIMINARIES The strength of fixed point theory lies in its extensive variety of applications.

Indeed, fixed point theory is a powerful tool to determine the existence and uniqueness theories of functional equations, integral equations, matrix equations, ordinary as well as partial differential equations, random differential equations, variational inequalities, etc, besides facilitating various problems arising in different domains, such as approximation theory, differential geometry, eigenvalue problems, functional analysis, operator theory and topology. Additionally, fixed point theory deals with various mathematical models representing phenomena arising in chemical equations, control theory, electrical heating in Joule-Thomson effect, equilibrium points in economy, fluid flow, fractal theory, game theory, Nash equilibrium, neutron transport theory, optimization theory, potential theory, probability theory, steady state temperature distribution, etc. Moreover, many practical and research problems in various fields beyond mathematics can be reduced to fixed point problems, which include biology, chemistry, computer science, economics, engineering, global analysis statistics, operations research, physics, etc. Though the existence or non existence of a fixed point is an intrinsic property of a mapping, there do exist many necessary or sufficient conditions for the existence of fixed points involving a mixture of topological, order-theoretic or geometric properties on the mapping or on its domain. Historically, the motivation of fixed point theory can be traced back to the mathematical activities of great mathematicians, such as Cauchy [46], Liouville [111], Lipschitz [112], Peano [137], Picard [138], Poincaré [141] and some others. In any case, formally, the start of the twentieth century marks the initiation of the topological fixed point theory with the forerunner work of the Dutch mathematician Brouwer [43]. Indeed, Brouwer [43] and Hadamard [74], independently, proved a fixed point result for a sphere whose extension to n-dimensional Euclidean space is one of the earliest result which is popularly referred as Brouwer Fixed Point Theorem: Theorem 1.1.1. [43, 74] A continuous self-mapping on a closed unit ball in Rn has at least one fixed point.

C HAPTER 1: I NTRODUCTION AND P RELIMINARIES

3

Interestingly, Theorem 1.1.1 for n = 1 reduces to a well known result of calculus refers as Intermediate Value Theorem which ensures a fixed point for every continuous self-mapping on a closed interval. Brouwer’s theorem remains one of the key theorems characterizing the topology of Euclidean spaces which earned a place to this theorem among the fundamental theorems of topology. In this regard, the survey article entitled "Ninety years of the Brouwer fixed point theorem" of Park [134] deserves a special mention. Two decades later, Schauder [161] extended Theorem 1.1.1 to an infinite dimensional space as under: Theorem 1.1.2. [161] A continuous self-mapping on a compact convex subset of a Banach space has a fixed point. Thereafter, in 1935, Tychonoff [178] extended Theorem 1.1.2 to a locally convex topological vector space. Finally, it is worth mentioning here that Kakutani [178] proved a generalization of Theorem 1.1.1 to multi-functions while Bohnenblust and Karlin [40] gave the multi-valued analogue of Theorem 1.1.2.

The current chapter, as usual, is elementary in nature wherein some fundamental notions, definitions and elementary results (needed in forthcoming chapters) are presented. Of course, a postgraduate level knowledge is supposed and therefore not included in. In fact, it is never possible to give a complete description of a wide subject like fixed point theory in a few pages. However, for a comprehensive study of fixed point theory and its related results, the reader can refer to classical books of Agarwal et al. [9], Carl and Heikkilä [45], Dugundji and Granas [64], Goebel and Kirk [71], Gopal et al. [73], Khamsi and Kirk [103], Kirk and Sims [106], Rus [157], Singh et al. [173] and Smart [174].

4


1.2

Metric fixed point theory

Metric fixed point theory is comprised of such fixed point results wherein geometric properties on the ambient space and/or underlying mapping are effectively utilized. This type of research is relatively not new but still a contemporary area of research. Although a substantial number of definitive results have already been discovered, many others are still open. There are many questions awaiting answers regarding the limits to which the theory may be extended. Some of these questions are merely tantalizing while others suggest substantial new avenues of research. The first natural fixed point result was in 1922 by the Polish mathematician Banach [31]. This result is known in the literature as Banach contraction principle and remains the most versatile elementary results in metric-theoretical fixed point theory. Definition 1.2.1. [31] Let ( M, d) be a metric space. A mapping S : M → M is said to be contraction if there is λ ∈ [0, 1) such that d(Su, Sv) ≤ λd(u, v), ∀ u, v ∈ M.

(1.2)

Theorem 1.2.1. [31] Let ( M, d) be a complete metric space and S : M → M a contraction mapping. Then S has a unique fixed point. Furthermore, Banach contraction principle guarantees that Picard sequence of S based at any point converges to the fixed point, i.e., starting at any point u0 ∈ M, the repeated iterations of the mapping at u0 yields a sequence that converges to the unique fixed point of S. The advantage of this principle is that its hypothesis is very simple and always gives a unique fixed point which can be found using a straightforward method. The only disadvantage attached to this principle is that assuming the mapping to be contraction forces the mapping S to be continuous at each point of the space. However, this principle is widely considered as the source of metric-theoretical fixed point theory and one of the most fundamental and powerful tools of nonlinear analysis.


5

Due to its wide range of fruitful applications within as well as outside mathematics, this principle has undergone a number of extensions and generalizations in the last ninety years and still good enough to yield new generalizations. To mention a few, we can refer to Alber and Guerre-Delabriere [16], Boyd and Wong [41], Browder [44], Chatterjea [48], C’irić [53, 54], Geraghty [69], Hardy and Rogers [76], Kannan [98, 99], Khojasteh et al. [105], Matkowski [115], Meir and Keeler [118], Popa [142], Reich [151], Rhoades [154], Samet et al. [159] and Wardowski [187]. For comprehensive studies, readers can refer to Jachymski [86], Park [133] and Rhoades [152, 153, 155].

In what follows, we shed light on some of the generalizations of this celebrated principle which we are going to utilize in subsequent chapters.

1.2.1

Certain Generalized Linear Contractions

Recall that for a metric space ( M, d) and a mapping S : M → M, Banach contraction principle employs the elements u, v, Su and Sv where u, v ∈ M in a concise manner via the components d(u, v) and (Su, Sv) only. Intuitively, the question arises: whether we are able to obtain similar conclusion under different or more components in the contraction condition? Many authors answered this question positively in different ways. In the following few lines we present only selected ones.

Kannan [98] introduced the following contractive condition: there exists λ ∈

(0, 1/2) such that d(Su, Sv) ≤ λ d(u, Su) + d(v, Sv) , ∀u, v ∈ M

(1.3)

and proved a fixed point theorem using contraction condition (1.3) rather than (1.2). Interestingly, both conditions are independent (see [99]). An advantage of Kannan contraction over Banach’s is that the condition (1.3) does not force the mapping S to be continuous on the whole space.

6

C HAPTER 1: I NTRODUCTION AND P RELIMINARIES In 1971, Reich [151] generalized both Banach and Kannan fixed point theorems

using the following contractive condition: d(Su, Sv) ≤ αd(u, v) + βd(u, Su) + γd(v, Sv), ∀u, v ∈ M,

(1.4)

where α, β and γ are nonnegative reals with α + β + γ < 1. Later on, C’irić [53] generalized the conditions (1.2), (1.3) and (1.4) using the following contractive condition: d(Su, Sv) ≤ λα(u, v)d(u, v) + β(u, v)d(u, Su) + γ(u, v)d(v, Sv) h i + δ(u, v) d(u, Sv) + d(v, Su) , ∀u, v ∈ M, (1.5) where α, β, γ, δ : M × M → [0, 1) are such that sup{α(u, v) + β(u, v) + γ(u, v) + 2δ(u, v)} < 1. Rhoades [155] observed that condition (1.5) implies the following condition due to C’irić [54]: d(Su, Sv) ≤ λ max MS (u, v), ∀u, v ∈ M,

(1.6)

where λ ∈ [0, 1) and ( MS (u, v) =

) d(u, Sv) + d(v, Su) d(u, v), d(u, Su), d(v, Sv), . 2

(1.7)

Usually, a mappings satisfying condition (1.6) are referred in the literature as generalized or C’irić-type contractions. In this dissertation, we adopt the first one and from now on, we use MS (u, v) for the expression (1.7). Further, the following notions will be adopted wherein S, T : M → M and u, v ∈ M: ( ) d(u, Su) + d(v, Sv) d(u, Sv) + d(v, Su) mS (u, v) = max d(u, v), , . 2 2 ( ) d( Tu, Sv) + d( Tv, Su) . MS,T (u, v) = d( Tu, Tv), d( Tu, Su), d( Tv, Sv), 2 ( mS,T (u, v) =

) d( Tu, Su) + d( Tv, Sv) d( Tu, Sv) + d( Tv, Su) d( Tu, Tv), , . 2 2


7

In 2012, Samet et al. [159] introduced the concept of α-admissible mapping and established various fixed point theorems for such mappings defined on complete metric spaces: Definition 1.2.2. [159] Let M be a non-empty set and α : M × M → [0, ∞). A mapping S : M → M is said to be α-admissible if α(u, v) ≥ 1 ⇒ α(Su, Sv) ≥ 1, ∀u, v ∈ M. The following remains a simplified form of a result due to Samet et al. [159]: Theorem 1.2.2. [159] Let ( M, d) be a complete metric space and S : M → M. Suppose that the following conditions hold: (i) there exists λ ∈ [0, 1) such that α(u, v)d(Su, Sv) ≤ λd(u, v), ∀u, v ∈ M, (ii) S is α-admissible, (iii) there exists u0 ∈ M such that α(u0 , Suo ) ≥ 1, (iv) S is continuous. Then S has a fixed point. Indeed, Theorem 1.2.2 was proved employing a nondecreasing function ψ : n n [0, ∞) → [0, ∞) such that ∑∞ n=1 ψ ( t ) < ∞ for each t > 0 where ψ is the n-the

iterates of ψ, but for simplicity, we choose ψ(t) = λt for λ ∈ [0, 1). This leads us to another type of the generalizations of Banach contraction principle to be summarized in the following subsection.

1.2.2

Nonlinear Contractions

In order to extend Banach Contraction Principle, several researchers attempted to absorb the contraction constant λ in the inequality (1.2) by a mapping ϕ : [0, ∞) →

[0, ∞) satisfying ϕ(t) < t, for all t > 0. Such contractions are, often, called nonlinear contractions wherein the adjective “nonlinear" has the meaning “not necessarily linear" and the utilized function “ϕ" is referred as a control function. The first ever such attempt was made by Browder [44].

8


Theorem 1.2.3. [44] Let ( M, d) be a complete metric space and S : M → M. If there exists an increasing right continuous control function ϕ such that d(Su, Sv) ≤ ϕ(d(u, v)), ∀u, v ∈ M, then S has a unique fixed point. Later on, many researchers generalized Banach contraction principle by varying the properties of the function ϕ. For such types of control functions, we refer the reader to [14] and references therein. In this presentation, we utilize control functions due to Boyd and Wong [41] and Matkowski [115] in Chapters 2 and 3 respectively. Boyd-Wong Contractions: Boyd and Wong [41] observed that the monotonicity in Theorem 1.2.3 is not necessary and merely the right-upper semi-continuity on ϕ will serve the purpose. Based on this observation, the authors suggested the following family of control functions which remains a larger class as compared to the class of Browder control functions. Definition 1.2.3. [41] A control function ϕ is said to be a Boyd-Wong (in short BW) if it is upper semi-continuous from the right. Matkowski Contractions: In 1975, Matkowski [115] initiated a special type of nonlinear contractions wherein the following control function was utilized: Definition 1.2.4. [115] A function ϕ : [0, ∞) → [0, ∞) is said to be a Matkowski (control) function if it satisfies the following conditions: (i) ϕ is increasing in [0, ∞), (ii) lim ϕn (t) = 0, ∀t > 0. n→∞

Observe that Matkowski function is often referred as comparison function.


1.2.3

9

Weak Contractions:

The following definition is needed in our subsequent discussions: Definition 1.2.5. [104] A function ψ : [0, ∞) → [0, ∞) is said to be an altering distance function if it is continuous, increasing and satisfies ψ(t) = 0 if and only if t = 0. In 1997, Alber and Guerre-Delabriere [16] introduced the notion of weak contraction which is often called ϕ-weak contractions as under: Definition 1.2.6. [16] Let ( M, d) be a metric space. A mapping S : M → M is said to be ϕ-weakly contractive mapping if d(Su, Sv) ≤ d(u, v) − ϕ(d(u, v)), ∀u, v ∈ M,

(1.8)

where ϕ is an altering distance function. Further, the authors of [16] utilized the idea of weak contractions to prove the existence and uniqueness of a fixed point for a self-mapping on Hilbert spaces. In 2001, Rhoades [154] showed that the main result contained in [16] remains true on complete metric spaces too. A detailed discussion on ϕ-weak contractions forms the content of Chapter 4.

1.2.4

Unified Fixed points Results

As mentioned earlier, a multitude of contraction conditions enriching Banach contraction principle have been introduced and utilized wherein every new contraction gives rise a relatively new theorem which requires an independent proof as well. Nowadays, there is a tradition of proving unified fixed point results employing an auxiliary function general enough to yield several contractions and henceforth several fixed point results in one go. In the following lines, we collect three of such auxiliary functions which we are going to utilize in Chapters 5, 6 and 7 respectively.

10

C HAPTER 1: I NTRODUCTION AND P RELIMINARIES In 1997, Popa [142] initiated the idea of unifying several contraction conditions

in one go. To accomplish this, he introduced the idea of the following implicit function: Definition 1.2.7. Consider the family I of all continuous functions I : [0, ∞)6 → R. In the respect of the family I , consider the following conditions: (I1 ) I is decreasing in the fifth and sixth variables, (I2 ) there exists h ∈ (0, 1) such that for u, v ≥ 0 with (I2a ) I (u, v, v, u, u + v, 0) ≤ 0 or (I2b ) I (u, v, u, v, 0, u + v) ≤ 0 implies u ≤ hv, (I3 ) I (u, u, 0, 0, u, u) > 0, ∀u > 0. Observe that on setting I (t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − λt2 , λ ∈ (0, 1), Banach contraction condition can be deduced if we assume that I (t1 , t2 , t3 , t4 , t5 , t6 ) ≤ 0 as t1 = d(Su, Sv) and t2 = d(u, v). Similarly, suitable selections of the function I yield several contraction conditions. In recent years, the idea of implicit function has been utilized by several authors and by now, there exists a considerable literature on this theme (see [10, 17–20, 22, 37, 81, 83, 84, 144–147, 184]).

A very recent and noted attempt to extend Banach contraction principle was due to Wardowski [187] wherein the author initiated the idea of F-contraction: Definition 1.2.8. [187] Let ( M, d) be a metric space. A mapping S : M → M is said to be an F-contraction if there exists τ > 0 such that d(Su, Sv) > 0 ⇒ τ + F (d(Su, Sv)) ≤ F (d(u, v)), ∀u, v ∈ M, where F : (0, ∞) → R is a mapping satisfying the following conditions: F1: F is strictly increasing, F2: for every sequence {sn } of positive real numbers, lim sn = 0 ⇔ lim F (sn ) = −∞,

n→∞

n→∞

F3: there exists k ∈ (0, 1) such that lim

s →0+

sk F (s)

= 0.


11

Some well known members of F are F (s) = ln s, F (s) = s + ln s, F (s) =

− √1 s

and F (s) = ln(s2 + s). Moreover, Wardowski [187] proved that every F-contraction mapping on a complete metric space possesses a unique fixed point. Further, on varying the elements of F suitably, a variety of known contractions in the literature can be deduced. For example, Banach contraction principle can be deduced if we choose F (s) = ln s. Evidently, the idea of F-contraction has attracted the attention of several researchers and by now, there exists a considerable literature in enriching this concept (e.g., [6, 21, 32, 35, 80, 102, 110, 119, 132, 135, 139, 140, 163, 164, 170–172, 179, 185, 188]).

Last but not least, Khojasteh et al. [105] introduced the idea of simulation function which is also designed to unify several contractions. Definition 1.2.9. [105] A mapping ξ : [0, ∞) × [0, ∞) → R is said to be a simulation function if it satisfies the following conditions:

(ξ1) ξ (0, 0) = 0, (ξ2) ξ (y, x ) < x − y, ∀y, x > 0, (ξ3) if { xn } and {yn } are sequences in (0, ∞) such that lim xn = lim yn > 0, then n→∞

n→∞

lim sup ξ ( xn , yn ) < 0. n→∞

Definition 1.2.10. [105] A self-mapping S on a metric space ( M, d) is said to be Z contraction with respect to a simulation function ξ if the following condition is satisfied: ξ (d(Su, Sv), d(u, v)) ≥ 0, ∀u, v ∈ M.

(1.9)

Observe that Banach contraction principle can be derived on choosing ξ (y, x ) = x − λy, where λ ∈ [0, 1). Likewise, several other contractions can also be derived choosing ξ suitably. For further work on simulation functions, one can consult [24, 100, 126, 156] and some others.

12


1.2.5

Theory of Coincidence Points

Recall that an element u ∈ M is said to be a fixed point of a self-mapping S on M if Su = u. This last equation can be written as Su = I M u. This observation derives the following expected question: under what conditions one can replace the identity mapping by another self-mapping T on M such that Su = Tu? The answer to this question opened a new door towards a new type of activities in fixed point theory under the umbrella of theory of coincidence points. Definition 1.2.11. Let M be a non-empty set and S, T : M → M. Also, let u, u ∈ M be such that Su = Tu = u. Then u is said to be a coincidence point of the pair (S, T ) and u is said to be the point of coincidence of the pair. Further, if u = u, then u is said to be a common fixed point of the pair (S, T ). The earliest metrical coincidence theorem was due to Machuca [114] in 1967. One year later, Goebel [70] derived a sharpened version of the main result of Machuca [114] which, for the sake of simplicity, is particularized as under: Theorem 1.2.4. [70] Let ( M, d) be a metric space and S, T : M → M. Suppose that the following conditions hold: (i) S( M) ⊆ T ( M), (ii) there exists λ ∈ [0, 1) such that d(Su, Sv) ≤ λd( Tu, Tv), ∀ u, v ∈ M, (iii) one of S( M) and T ( M ) is a complete subspace of M. Then the pair (S, T ) has a coincidence point. A clear observation on Theorem 1.2.4 is that the mappings of the pair (S, T ) are not commuting. In order to investigate the existence of a common fixed point, the following question due to Isbell [85] remains pertinent and crucial: Problem 1.2.1. [85] Let (S, T ) be a pair of two commuting continuous self-mappings on the unit interval. Do they have a common fixed point?


13

The answer to Problem 1.2.1 was settled in negative as Baxter [33] constructed a pair of commuting mappings with no common fixed point employing a limiting process. Thus, in order to coin a common fixed point theorem, one is required to impose extra conditions either on the space or on the involved mappings which is evident from the existing literature on common fixed point theory. In 1976, Jungck [95] generalized Banach contraction principle by proving a common fixed point theorem for commuting mappings by employing a constructive procedure of sequence of joint iteration. Theorem 1.2.5. [95] Let( M, d) be a complete metric space and S, T : M → M. Suppose that the following conditions hold: (i) S( M) ⊆ T ( M), (ii) there exists λ ∈ [0, 1) such that d(Su, Sv) ≤ λd( Tu, Tv), ∀ u, v ∈ M, (iii) T is continuous, (iv) S and T are commuting mappings. Then the pair (S, T ) has a unique common fixed point. In 1982, Sessa [166] introduced a weaker notion of the commutativity condition (given in in Theorem 1.2.5) which runs as follows: Definition 1.2.12. [166] Let ( M, d) be a metric space and S, T : M → M. We say that S and T are weakly commuting mappings if d(STu, TSu) ≤ d(Su, Tu), ∀ u ∈ M. Four years later, Jungck [95] weakened the notion of weak commutativity by defining a compatible pair of mappings. Definition 1.2.13. [95] Let ( M, d) be a metric space and S, T : M → M. We say that

(S, T ) is a compatible pair of mappings if for any sequence {un } ⊆ M, lim S(un ) = lim Tun = w ⇒ lim d(STun , TSun ) = 0, w ∈ M.

n→∞

n→∞

n→∞

14


Definition 1.2.14. [96] Let ( M, d) be a metric space and S, T : M → M. We say that S and T are weakly compatible mappings if the pair (S, T ) commutes on the set of coincidence points. Lemma 1.2.1. [14] Let ( M, d) be a metric space and S, T : M → M a pair of weakly compatible mappings. Then every point of coincidence of the pair (S, T ) remains a coincidence point. In subsequent years, various researchers of the domain have been studying a host of weaker compatibility conditions and utilized the same to develop common fixed point theorems. The comprehensive and lucid collections of such conditions and their interplay can be found in Agarwal et al. [7], Kadelburg et al. [97] and Murthy [122].

1.3

Order-Theoretic fixed point theory

After the appearance of the notion of metric space by Fréchet [68], many researchers attempted to improve this central idea by defining various related concepts using varying views and ideas. One such important notion is that of an ordered metric space. Although such spaces are in fashion only recently, yet these spaces were introduced and studied earlier, e.g., Knaster [107], Tarski [175], Vandergraft [181] and Wolk [189]. Evidently, proving order-theoretic fixed point results has attracted the attention of several researchers and by now, there exists considerable literature enriching this concept (e.g.,[8,11,13–15,28,47,52,57,89–91,109,124,130,168,169,180]). Recall that a binary relation on a non-empty set M is said to be a partial order if it is reflexive, antisymmetric and transitive. A set M together with a partial order

is said to be an ordered set and often denoted by ( M, ). If ( M, d) is a metric space and is a partial order (in short, order) on M, then the triplet ( M, d, ) refers to this partial order metric space. The statement u v is read as “u precedes v”. Analogously, we also write u v which is read as “u succeeds v". Further, we say that u and v are comparable if either u v or u v. For brevity, we denote it by u ≺ v.


15

In order-theoretic fixed point theory, we study those fixed point results, in which the ambient space is equipped with a partial order and the underlying mapping remains order-preserving. An ordered set M is said to have the fixed point property if every order-preserving mapping on M admits a fixed point. Remark 1.3.1. In the framework of ordered metric spaces, the contraction inequalities (such as (1.2)) are merely required to hold only on comparable elements. In 1989, Sun and Sun [92] initiated ordered fixed point theory while, in 1996, Wanka [186] published a paper concerning approximation theory in ordered spaces. Inspired by earlier mentioned core results, many authors produced remarkable results in this direction. For the work of this kind, one can be referred to Abian ¨ [4], Abian and Brown [5], Amann [23], Bjorner [39], Brondsted [42], De Marr [63], Dugundji and Granas [64], Kurepa [108], Pasini [136], Vandergraft [181] and Wong [190]. Applications of order-theoretic fixed point theory are scattered in a wide range of diverse fields, e.g., multi-valued nonlocal and/ or discontinuous partial differential equations of elliptic and parabolic type, differential equations and integral equations besides being effectively used in mathematical economics and game theory.

1.3.1

Classical Order-Theoretic Notions

As the main goal of this dissertation is proving order-theoretic fixed point results, in this subsection, we compile basic order-theoretic notions and definitions needed in our subsequent chapters. Definition 1.3.1. [113] Let ( M, ) be an ordered set and E ⊆ M. Then • the subset E is said to be a totally ordered set if every pair of elements of E are comparable, • an element u0 ∈ M is said to be an upper bound of E if u0 succeeds every element of E, i.e., u u0 , ∀u ∈ E, • an element l ∈ M is said to be a lower bound of E if l precedes every element of E, i.e., u l, ∀u ∈ E.

16


Definition 1.3.2. [58] Let ( M, ) be an ordered set and S, T : M → M. Then (i) S is said to be T-increasing if Tu Tv ⇒ Su Sv, for any u, v ∈ M, (ii) S is said to be T-decreasing if Tu Tv ⇒ Su Sv, for any u, v ∈ M, (iii) S is said to be T-monotone if it is either T-increasing or T-decreasing. On setting T = I M in part (i) of Definition 1.3.2, S is said to be an increasing mapping (similarly for the other parts). Definition 1.3.3. [13] Let ( M, ) be an ordered set, E ⊆ M and u, v ∈ E. A finite subset

{e1 , e2 , ..., ek } of E is said to be ≺-chain between u and v in E if (i) k ≥ 2, (ii) e1 = u and ek = v, (iii) ei ≺ ei+1 for every i (1 ≤ i ≤ k − 1). We denote by C(u, v, ≺, E) the class of all ≺-chains between u and v in E. In particular for E = M, we write C(u, v, ≺) instead of C(u, v, ≺, M). Definition 1.3.4. [12] Let ( M, ) be an ordered set. A sequence {un } ⊆ M is said to be term-wise monotone if consecutive terms of {un } are comparable. Moreover, M is said to be sequentially chainable if the range of every term-wise monotone sequence in M remains a totally ordered subset of M. Clearly, if ( M, ) is sequentially chainable, then for every term-wise monotone sequence {un } in M, we have un ≺ um , ∀n, m ∈ N. Definition 1.3.5. [13] Let ( M, ) be an ordered set and S, T : M → M. Then ( M, ) is said to be (S, T )-directed if for every pair u, v ∈ M, there exists w ∈ M such that Tu ≺ Sw and Tv ≺ Sw. Particularly, for T = I M , ( M, ) is said to be S-directed and if S = T = I M , it is called directed set. Definition 1.3.6. [12] An ordered metric space ( M, d, ) is said to enjoy TCC (term-wise monotone-convergence-c-bound) property if every term-wise monotone convergent sequence

{un } in M admits a subsequence, which is term-wise bounded by the limit of {un } as a c-bound.


17

Notice that Definition 1.3.6 is formulated using a property used within assumption (6) in Theorem 4 of Nieto and Rodríguez-López [128]. Definition 1.3.7. [15] An ordered metric space ( M, d, ) is said to be (i) O-complete if every increasing Cauchy sequence converges in M, (ii) O-complete if every decreasing Cauchy sequence converges in M, (iii) O-complete if every monotone Cauchy sequence converges in M. Remark 1.3.2. In an ordered metric space, completeness ⇒ O-completeness ⇒ Ocompleteness (as well as O-completeness). Alam et al. [15] generalized the compatibility notion given in Definition 1.2.13 and utilized these relatively weaker notions to prove some common fixed point theorems. Definition 1.3.8. [15] Let ( M, d) be a metric space and S, T : M → M. Then the pair

(S, T ) is said to be (i) O-compatible, if for any sequence {un } ⊆ M and w ∈ M, Tun ↑ w and S(un ) ↑ w ⇒ lim d( TSun , STun ) = 0, n→∞

(ii) O-compatible, if for any sequence {un } ⊆ M and w ∈ M, Tun ↓ w and S(un ) ↓ w ⇒ lim d( TSun , STun ) = 0 and n→∞

(iii) O-compatible, if for any sequence {un } ⊆ M and w ∈ M , Tun ↑↓ w and S(un ) ↑↓ w ⇒ lim d( TSun , STun ) = 0. n→∞

Remark 1.3.3. In any metric space, commutativity ⇒ weak commutativity ⇒ compatibility ⇒ O-compatibility ⇒ O-compatibility (as well as O-compatibility) ⇒ weak compatibility. Definition 1.3.9. [160] Let ( M, d, ) be an ordered metric space and S, T : M → M. Then S is said to be T-continuous at u ∈ M if for every sequence {un } ⊆ M, d

d

Tun −→ Tu ⇒ S(un ) −→ Su. Moreover, S is said to be T-continuous if it is T-continuous at every point of M.

18

C HAPTER 1: I NTRODUCTION AND P RELIMINARIES Notice that particularly for T = I M , Definition 1.3.9 reduces to the usual

definition of the continuity. Definition 1.3.10. [15] Let ( M, d, ) be an ordered metric space and S, T : M → M. Then S is said to be (i) ( T, O)-continuous at u ∈ M if for every sequence {un } ⊆ M, d

Tun ↑ Tu ⇒ S(un ) −→ Su, (ii) ( T, O)-continuous at u ∈ M if for every sequence {un } ⊆ M, d

Tun ↓ Tu ⇒ S(un ) −→ Su and (iii) ( T, O)-continuous at u ∈ M if for every sequence {un } ⊆ M, d

Tun ↑↓ Tu ⇒ S(un ) −→ Su. Moreover, the mapping S is said to be ( T, O)- continuous if it is ( T, O)-continuous at every point of M (similar argument holds for other cases). Notice that on setting T = I M in part (i) of Definition 1.3.10, S is said to be O-continuous (similar argument holds for other cases). Remark 1.3.4. In an ordered metric space, T-continuity ⇒ ( T, O)-continuity ⇒ ( T, O)continuity (as well as ( T, O)-continuity). Definition 1.3.11. [14] Let ( M, d, ) be an ordered metric space and S, T : M → M. Then (i) ( M, d, ) enjoys T-ICU property if T-image of every increasing convergent sequence

{un } in M is bounded above by T-image of its limit (as an upper bound), i.e., un ↑ x ⇒ Tun Tu, ∀ n ∈ N, (ii) ( M, d, ) enjoys T-DCL property if T-image of every decreasing convergent sequence

{un } in M is bounded below by T-image of its limit (as a lower bound), i.e., un ↓ u ⇒ Tun Tu, ∀ n ∈ N, (iii) ( M, d, ) enjoys T-MCB property if it has both T-DCL as well as T-DCL property. The following notions are relatively weaker than those of Definition 1.3.11.


19

Definition 1.3.12. [15] Let ( M, d, ) be an ordered metric space and S, T : M → M. Then (i) ( M, d, ) enjoys T-ICC property if every increasing convergent sequence {un } in M has a subsequence {unk } such that every term of { Tunk } is comparable with T-image of the limit of {un }, i.e., un ↑ u ⇒ there exists a subsequence {unk } of {un } with Tunk ≺ Tu, ∀ k ∈ N, (ii) ( M, d, ) enjoys T-DCC property if every decreasing convergent sequence {un } in M has a subsequence {unk } such that every term of { Tunk } is comparable with T-image of the limit of {un }, i.e., un ↓ u ⇒ there exists a subsequence {unk } of {un } with Tunk ≺ Tu, ∀ k ∈ N and (iii) ( M, d, ) enjoys T-MCC property if every monotone convergent sequence {un } in M has a subsequence {unk } such that every term of { Tunk } is comparable with T-image of the limit of {un }, i.e., un ↑↓ u ⇒ there exists a subsequence {unk } of

{un } with Tunk ≺ Tu, ∀ k ∈ N. Notice that under the restriction T = I M , ( M, d, ) in Definition 1.3.12(i) is said enjoy ICC property (similar argument holds for other cases ). Remark 1.3.5. For an ordered metric space, T-DCL property ⇒ T-ICC property. T-DCL property ⇒ T-DCC property. T-MCB property ⇒ T-MCC property. T-MCC property ⇒ T-ICC property (as well as T-DCC property).

1.3.2

Order-Theoretic Fixed Point Results

In 2004, Ran and Reurings [150] proved an order-theoretic analogue of Banach contraction principle which marks the beginning of a vigorous research activity. This result was discovered while investigating the solutions to some special matrix equations. However, the origin of this result can be traced back to Turinici [176, 177]. In continuation of Ran and Reurings, Nieto and Rodríguez-López [127, 128] proved two very useful results and used them to solve some differential equations. Later ,

20


many mathematicians has shown keen interest in the investigations of the metric fixed point problems for monotone mappings defined in a partially ordered metric space. For a fresh review of this type of research, we refer to Bachar and Khamsi [29] and references cited therein.

Based on the main goal of this dissertation, in what follows, we collect relevant basic results needed in our subsequent discussion. We begin with Ran and Reurings’ result which runs as follows: Theorem 1.3.1. [150] Let ( M, d, ) be an ordered metric space and S : M → M. Suppose that the following conditions hold: (i) ( M, d) is complete, (ii) S is monotone, (iii) S is continuous, (iv) there exists u0 ∈ M such that u0 ≺ S(u0 ), (v) there exists λ ∈ [0, 1) such that d(Su, Sv) ≤ λd(u, v), ∀u, v ∈ M with u v. Then S has a fixed point. Ran and Reurings, also, suggested the following condition to ensure the uniqueness of the fixed point concerning Theorem 1.3.1: Every pair of elements of M has a lower bound and an upper bound.

(1.10)

Furthermore, Ran and Reurings applied Theorem 1.3.1 to obtain a unique positive definite solutions of the following linear matrix equations M − A1∗ MA1 − ... − ..A∗m MAm = Q and M + A1∗ MA1 + ... + A∗m MAm = Q; where Q is a positive definite matrix, M is an unknown matrix and A1 , ..., Am are arbitrary n × n matrices.


21

In attempt to prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution, the Spanish mathematicians Nieto and RodríguezLópez [127], in 2005, proved an appropriate fixed point theorem in partially ordered sets as follows: Theorem 1.3.2. [127] Let ( M, d, ) be an ordered metric space and S : M → M. Suppose that the following conditions hold: (i) ( M, d) is complete, (ii) S is increasing, (iii) either S is continuous or ( M, d, ) enjoys ICU property, (iv) there exists u0 ∈ M such that u0 S(u0 ), (v) there exists λ ∈ [0, 1) such that d(Su, Sv) ≤ λd(u, v), ∀ u, v ∈ M with u v. Then S has a fixed point. Observe that Theorem 1.3.2 remains an improved version of Theorem 1.3.1. Two years later, Nieto and Rodríguez-López [128] considered decreasing mappings and proved another version of Theorem 1.3.2 as under: Theorem 1.3.3. [128] Theorem 1.3.2 remains true if the conditions (iii), (iii) and (iv) are replaced by the conditions (ii)0 , (iii)0 and (iv)0 respectively: (iii)0 S is decreasing, (iii)0 either S is continuous or ( M, d, ) enjoys DCU property, (iv)0 there exists u0 ∈ M such that u0 ≺ S(u0 ). Additionally, Nieto and Rodríguez-López [128] obtained the uniqueness of the fixed point and consequent global convergence of the method of successive approximations by adding the following hypothesis which remains relatively weaker than condition (1.10) and equivalent to saying that the set M is directed.

Every pair of elements of M has a lower bound or an upper bound.

(1.11)

22

C HAPTER 1: I NTRODUCTION AND P RELIMINARIES The following lemmas are crucial for the proofs of our results. We begin with

the following lemma which is contained in the proof of Theorem 2 in [77]. Lemma 1.3.1. Let (M,d) be a metric space and {un } a sequence in M such that lim d(un , un+1 ) = 0. If {un } is not a Cauchy sequence, then there exist e > 0 and two

n→∞

subsequences {unk } and {umk } of {un } such that nk > mk ≥ k, such that d(umk , unk ) ≥ e, d(umk , unk −1 ) < e and the following four sequences tend to e when k → ∞: d(umk , unk ), d(umk +1 , unk ), d(umk , unk +1 ) and d(umk +1 , unk +1 ). Lemma 1.3.2. [14] Let ( M, ) be an ordered set and S, T : M → M. If S is T-monotone and Tu = Tv, then Su = Sv. Lemma 1.3.3. [75] Let M be a non-empty set and S : M → M. Then there exists a subset E ⊆ M such that S( E) = S( M) and S : E → M is one-one.

CHAPTER

2

Results for Boyd-Wong Type Contractions

In this chapter, we prove some order-theoretic fixed point theorems for a monotone generalized contraction wherein the utilized control function is of BoydWong type. The newly obtained results generalize several known results of the ´ c et al. [58], existing literature especially those contained in Agarwal et al. [8], Ciri´ Nieto and Rodríguez-López [127], Ran and Reurings [150] and Wu and Liu [191]. We furnish several examples to illustrate the usability of our results especially one of these examples disproves one of the remarks contained in Agarwal et al. [8]. As an application, we prove a fixed point result for a mapping satisfying Boyd-Wong integral type contraction in ordered metric space.

2.1

Introduction and Preliminaries

Recall that one of the natural extensions of Banach contraction principle is obtained by enlarging the class of mappings employing non-linear contractions wherein a suitable control function is used so as to merge contraction constant (see Subsection 1.2.2). One of the most widely and early used control functions are essentially due to Boyd and Wong [41] wherein the authoress utilized an upper semi-continuous The content of this chapter is based on the following article: M. Imdad and R. Gubran Ordered-theoretic fixed point results for monotone generalized Boyd-Wong and Matkowski type contractions J. Adv. Math. Stud., 10 (1) 2017, 49-61.

23

24

C HAPTER 2: R ESULTS FOR B OYD -W ONG T YPE C ONTRACTIONS

from the right (see Definition 1.2.3). Employing this control function, Boyd and Wong [41] generalized Banach contraction principle as follows: Theorem 2.1.1. [41] Let ( M, d) be a complete metric space and S : M → M. If there exists a Boyd-Wong function ϕ such that d(Su, Sv) ≤ ϕ(d(u, v)), ∀u, v ∈ M, then S has a unique fixed point. In 2013 Wu and Liu [191] proved the following result which remains an ordertheoretic analogue of Theorem 2.1.1. Theorem 2.1.2. [191] Let ( M, d, ) be a complete ordered metric space and S : M → M an increasing mapping. Suppose that the following conditions hold: (a) there exists u0 ∈ M such that u0 Su0 , (b) either S is continuous or ( M, d, ) enjoys ICU property, (c) there exists a Boyd-Wong function ϕ such that d(Su, Sv) ≤ ϕ(d(u, v)), ∀u, v ∈ M with u v. Then S has a fixed point. The following result is needed in our subsequent discussion: Lemma 2.1.1. [15] Let ϕ be a Boyd-Wong function. If { an } ⊂ (0, ∞) is a sequence such that an+1 ≤ ϕ( an ) ∀ n ∈ N, then lim an = 0. n→∞

2.2

Existence Results

Definition 2.2.1. Let ( M, d, ) be an ordered metric space. A mapping S : M → M is said to be a generalized Boyd-Wong contraction if there exists a Boyd-Wong function ϕ such that: d(Su, Sv) ≤ ϕ( MS (u, v)), ∀u, v ∈ M with u v.

(2.1)

Employing some recent order-theoretic notions especially those contained in Definitions 1.3.7, 1.3.10 and 1.3.12, we prove the following theorem for a generalized Boyd-Wong contraction:


25

Theorem 2.2.1. Let ( M, d, ) be an ordered metric space, E is a subspace of M and S : M → M such that S( M) ⊆ E. Moreover, suppose that the following conditions hold: (i) E is an O-complete subspace of M, (ii) S is an increasing mapping, (iii) there exists u0 ∈ M such that u0 Su0 , (iv) either S is O-continuous or ( E, d, ) enjoys ICC property, (v) S is generalized Boyd-Wong contraction. Then S has a fixed point. Proof. Consider u0 ∈ M such that u0 Su0 and define a sequence {un } in E by un := Sn u0 , ∀n ∈ N0 . As S is increasing, we have u0 u1 u2 u3 ... un .... Let dn := d(un+1 , un ). Observe that if dn = 0 for some n ∈ N0 , then un is a fixed point of S and we are through. We assert that: d n ≤ ϕ ( d n −1 ). To substantiate the assertion, on setting u = un and v = un−1 in (2.1), we have dn ≤ ϕ( MS (un , un−1 )), where

d ( u n , u n ) + d ( u n −1 , u n +1 ) MS (un , un−1 ) = max dn−1 , dn , dn−1 , 2 d ( u n −1 , u n +1 ) = max dn−1 , dn , . 2 On using the triangular inequality,

d(un−1 ,un+1 ) 2

≤

d n −1 + d n 2

≤ max {dn−1 , dn }. If

MS (un , un−1 ) = dn , then using the definition of Boyd-Wong function, we have dn ≤ ϕ(dn ) < dn which is a contradiction. Therefore, MS (un , un−1 ) = dn−1 and our assertion is established. Now, using lemma 2.1.1, we have lim dn = 0.

n→∞

Next, we show that {un } is a Cauchy sequence in E. If it is not so, then by

26


Lemma 1.3.1, there exist e > 0 and two sequences {unk }, {umk } ⊆ {un } such that nk > mk ≥ k, d(umk , unk ) ≥ e, d(unk −1 , umk ) < e and lim d(umk , unk ) = lim d(umk+1 , unk ) = lim d(umk , unk+1 )

k→∞

k→∞

k→∞

= lim d(umk+1 , unk+1 ) = e. (2.2) k→∞

Since nk > mk , on setting u = unk and v = umk in (2.1), we get d(unk +1 , umk +1 ) ≤ ϕ( MS (unk , um k )).

(2.3)

d ( u n k + 1 , u m k + 1 ) ≤ MS ( u n k , u m k ) ,

(2.4)

Therefore,

where MS (unk , um k ) = max d(unk , umk ), d(unk , unk +1 ), d(umk , umk +1 ), d ( u n k , u m k +1 ) + d ( u m k , u n k +1 ) . 2 d(unk ,umk +1 )+d(umk ,unk +1 ) Suppose MS (unk , um k ) = max d(unk , umk ), := tk . On using 2 (2.2) and taking the limit superior of (2.3) when k → ∞, we have e ≤ lim sup ϕ(tk ) k→∞

= lim sup ϕ(tk ), tk →e+

< e, which is a contradiction. Otherwise, if MS (unk , um k ) assumes any of the other two values, owing to (2.2), on taking the limit of (2.4) when k → ∞, we get e ≤ 0 which is a contradiction. Hence, {un } is a Cauchy sequence. Now, the O-completeness of E ensures the existence of some u ∈ E with un ↑ u. Observe that, if d(um , u) = 0 for some m ∈ N0 , then um+1 = Su which on letting m → ∞ gives rise u = Su so that we are done. Hence, in our further discussion, we assume d(u, un ) > 0, for all n ∈ N0 .


27

On using assumption (iv), if S is O-continuous, then u = lim un = lim Sun−1 = S lim un−1 = Su. n→∞

n→∞

n→∞

Otherwise, let ( E, d, ) enjoys ICC property. Then there exists a subsequence {unk } of {un } such that unk ≺ u, ∀ k ∈ N. Suppose d(u, Su) = a > 0. Then a ≤ d(u, Sunk ) + d(Sunk , Su)

≤ d(u, unk +1 ) + ϕ( MS (unk , u)). Hence, a < d(u, unk +1 ) + MS (unk , u),

(2.5)

where d(u, unk +1 ) + d(unk , Su) . MS (unk , u) = max d(unk , u), d(unk , unk +1 ), d(u, Su), 2

We distinguish the following two cases: If MS (unk , u) = max d(unk , u), d(unk , unk +1 ),

d(u,unk +1 )+d(unk ,Su) 2

, then taking the

limit on both the sides of (2.5) we arrives a contradiction in view of the fact that a > 0. In other case, (2.5) can be written as a ≤ d(u, unk+1 ) + ϕ( a), which on making k → ∞ on both the sides gives rise a ≤ ϕ( a), which is a contradiction. Thus, d(u, Su) = 0. This concludes the proof. Remark 2.2.1. Theorem 2.2.1 is a generalization of Theorems 1.3.1, 1.3.2 and 2.1.2 as well ´ c et al. [58]. as Corollary 2.7 of Ciri´ We furnish with the following example which creates a situation where Theorem 2.2.1 can be applied while Theorem 2.1.2 (due to Wu and Liu [191]) can not, even in the present of a linear control function.

28


Example 2.2.1. Consider M = {0, 1, 2} equipped with a metric d : M × M → R given by: d(u, v) =

  u + v, for u 6= v, for u = v,

 0, then ( M, d) is a complete metric space.

Consider a control function ϕ : [0, ∞) → [0, ∞) defined by ϕ(t) = t/3 and a self-mapping S on M defined by: Su =

  u − 1, for u 6= 0,  0,

for u = 0,

In order to verify the inequality (2.1) and due to the symmetricity of the metric, we distinguish the following three cases for u, v ∈ M: • If u = v, then d(Su, Sv) = 0. Thus, (2.1) holds trivially. Similarly if u = 0, v = 1. • If u = 0, v = 2, then d(S0, S2) = 1 and d(0, S2) + d(2, S0) ϕ( M(0, 2)) = ϕ max d(0, 2), d(0, S0), d(2, S2), 2 = ϕ(max {2, 0, 3, 3/2})

= 1. • If u = 1, v = 2, then d(S1, S2) = 1 and d(1, S2) + d(2, S1) ϕ( M(1, 2)) = ϕ max d(1, 2), d(1, S1), d(2, S2), 2 = ϕ(max {3, 1})

= 1. Hence, in all cases, inequality (2.1) holds so that Theorem 2.2.1 ensures the existence of the fixed point u = 0. Remark 2.2.2. The above example cannot be covered by Theorem 2.1.2 as d(S0, S2) is greater than ϕ(d(0, 2)).


29

On utilizing Definition 1.3.4, we close this section by proving a result for if the mapping S in Theorem 2.2.1 is supposed to be decreasing. Theorem 2.2.2. Let ( M, d, ) be an ordered metric space, E an subspace of M and S : M → M such that S( M ) ⊆ E. Moreover, suppose that the following conditions hold: (i) E is an O-complete subspace of M, (ii) S is a decreasing mapping, (iii) there exists u0 ∈ M such that u0 ≺ Su0 , (iv) either S is O-continuous or ( E, d, ) enjoys TCC property, (v) S is a generalized Boyd-Wong contraction, (vi) ( M, ) is sequentially chainable. Then S has a fixed point. Proof. Consider u0 such that u0 ≺ Su0 and define a sequence {un } in E by un := Sn u0 ∀n ∈ N. As S is decreasing mapping, {un } is termwise monotone, i.e., un ≺ un+1 , ∀n. Also, due to assumption (vi), for each m, n ∈ N, we have um ≺ un . Now, the proof is analogous to that of Theorem 2.2.1. Observe that, the TCC property of

( E, d, ) in this context plays the same role of ICC property in Theorem 2.2.1. Remark 2.2.3. Theorem 2.2.2 remains a generalization of Theorem 1.3.3

2.3

Uniqueness Results

After several failures in our efforts to prove corresponding uniqueness results, we are left with no option but to consider a relatively stronger contraction condition which is contained in the following definition. Definition 2.3.1. Let ( M, d, ) be an ordered metric space. A mapping S : M → M is said to be a lean generalized Boyd-Wong contraction, if: d(Su, Sv) ≤ ϕ(mS (u, v)), ∀u, v ∈ M with u v where ϕ is Boyd-Wong function.

(2.6)

30


Theorem 2.3.1. Let ( M, d, ) be an ordered metric space, E a subspace of M and S : M → M is such that S( M) ⊆ E. Moreover, suppose that the following conditions hold: (i) E is an O-complete subspace of M, (ii) S is an increasing mapping, (iii) there exists u0 ∈ M such that u0 Su0 , (iv) either S is O-continuous or ( E, d, ) enjoys ICC property, (v) S is a lean generalized Boyd-Wong contraction. Then S has a fixed point. Further, if ( M, ) is directed, then S has a unique fixed point. Proof. As every lean generalized Boyd-Wong contraction mapping is a generalized Boyd-Wong contraction, the existence of the fixed point of S follows from Theorem 2.2.1. To prove the uniqueness of such fixed point, let u, v ∈ M be two fixed points of S. We need to show that u = v. By the directedness of M, there exists some w ∈ M such that w is comparable to both u and v. Set w0 = w and define a sequence

{wn } in M by: wn := Sn w0 , ∀n ∈ N. As S is increasing and u ≺ w, we have u ≺ wn , ∀ n ∈ N. We claim that lim d(u, wn ) = 0.

n→∞

If d(u, wm ) = 0 for some m ∈ N, then we must have d(Su, wm+1 ) = 0, that is, d(u, wm+1 ) = 0. Consequently, d(u, wn ) = 0, ∀n ≥ m. Otherwise, setting u = u, v = wn in (2.6) gives rise d(u, wn+1 ) = d(Su, Swn )

≤ ϕ(mS (u, wn )), where

d(u, Su) + d(wn , wn+1 ) d(u, wn+1 ) + d(wn , Su) mS (u, wn ) = max d(u, wn ), , . 2 2 (2.7)


31

It is obvious that, d(wn , wn+1 ) ≤ d(u, wn+1 ) + d(wn , u). Thus, two cases arise. If d(u,Swn+1 )+d(wn ,Su) 2

≤ d(u, wn ), then from (2.7), we have d(u, wn+1 ) ≤ ϕ(d(u, wn ))

and since ϕ is Boyd-Wong function, the claim is established by Lemma 2.1.1 in this case. In the other case we have,

d(u,wn+1 )+d(wn ,u) 2

> d(u, wn ), that is, d(u, wn+1 ) > d(u, wn ).

On using the definition of ϕ, (2.7) can be written as d(u, wn+1 ) + d(wn , u) d(u, wn+1 ) ≤ ϕ 2 d(u, wn+1 ) + d(wn , u) < . 2 Hence, d(u, wn+1 ) < d(wn , u) which is a contradiction. Thus, in all cases, our claim is established. Similarly, one can show that lim d(v, wn ) = 0. n→∞

Finally, d(u, v) = d(u, wn ) + d(wn , v)

→ 0 as n → ∞ so that u = v. This concludes the proof. Remark 2.3.1. One can obtain dual type results corresponding to Theorems 2.2.1 and 2.3.1 if involved terms namely: O-completeness, O-continuity and the ICC property are, respectively, replaced by O-completeness, O-continuity and the DCC property provided assumption (iii) is replaced by the following: (iiia) there exists u0 ∈ M such that u0 Su0 . Remark 2.3.2. One can obtain similar results corresponding to Theorems 2.2.1 and 2.3.1 if involved terms namely: O-completeness, O-continuity and the ICC property are, respectively, replaced by O-completeness, O-continuity and the MCC property provided assumption (iii) is replaced by the following: (iiib) there exists u0 ∈ M such that u0 ≺ Su0 . Remark 2.3.3. The condition "( M, ) is directed" ensures the uniqueness of the fixed point in Theorem 2.2.2 provided condition (2.1) is replaced by condition (2.6).

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2.4

An Application

As an application of Theorem 2.2.1, we have the following fixed point result for a mapping satisfying Boyd-Wong integral type contraction in ordered metric space. Let Λ be the set of functions ω : [0, ∞) → [0, ∞) satisfying the following: (a) ω is Lebesgue-integrable mapping on each compact subset of [0, ∞); Re (b) 0 ω (t)dt > 0 ∀e > 0. Theorem 2.4.1. Let ( M, d, ) be an ordered metric space, E an O-complete subspace of M and S : M → M an increasing mapping such that S( M) ⊆ E. Suppose that for every u, v ∈ M with u v and ω ∈ Λ, we have d(Su, Sv) ≤

Z ϕ( M(u,v)) 0

ω (t)dt,

(2.8)

where ϕ is a Boyd-Wong function. If the assumptions (iii), (iv) and (v) of Theorem 2.2.1 are satisfied, then S has a fixed point. Proof. Define Γ : [0, ∞) → [0, ∞) by Γ(u) =

Ru 0

ω (t)dt, then Γ is a continuous

increasing function. Therefore, (2.8) can be written as d(Su, Sv) ≤ (Γ ◦ ϕ)( M (u, v)). The required follows directly from Theorem 2.2.1 if we show that Γ ◦ ϕ is a BoydWong function. Let λi be a sequence in [0, ∞) such that λi ↓ λ ≥ 0. Then lim sup(Γ ◦ ϕ)(λi ) = Γ lim sup ϕ(λi ) λi → λ +

λi → λ +

≤ (Γ ◦ ϕ)(λ). Thus, Γ ◦ ϕ is upper semi-continuous from the right. This concludes the proof.

CHAPTER

3

Results for Matkowski Type Contractions

The aim of this chapter is two-fold: Firstly, we prove some existence and uniqueness order-theoretic fixed point theorems for monotone generalized Matkowski contractions. As an application of our newly obtained results, we propose a result for a mapping satisfying Matkowski integral type contraction condition. Secondly, we prove some order-theoretic common fixed points results for T-increasing Matkowski contractions.

3.1


Recall that a Matkowski function (see Definition 1.2.4) is an increasing mapping ϕ : [0, ∞) → [0, ∞) satisfies lim ϕn (t) = 0, ∀t > 0. n→∞

The following observation highlights some basic proprieties of this function. Proposition 3.1.1. [115, 116] Let ϕ be a Matkowski function. Then (i) ϕ(t) < t, f or all t > 0. (ii) ϕ(0) = 0. The contents of this chapter are based on the following two articles: M. Imdad and R. Gubran Ordered-theoretic fixed point results for monotone generalized Boyd-Wong and Matkowski type contractions J. Adv. Math. Stud., 10 (1) 2017, 49-61. & R. Gubran, Idress A. Khan and M. Imdad Fixed point theorems for Matkowski-type nonlinear contractions in ordered metric spaces J. Inequal. Spec. Funct., 9 (3) 2018.

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34

C HAPTER 3: R ESULTS FOR M ATKOWSKI T YPE C ONTRACTIONS

Proof. (i) Assume there exists t0 > 0 with t0 ≤ ϕ(t0 ). Since ϕ is increasing, ϕ ( t0 ) ≤ ϕ2 ( t0 ), it follows that t0 ≤ ϕ2 ( t0 ) so that, by induction, t0 ≤ ϕn (t0 ), ∀n ∈ N. On letting n → ∞, we get t0 ≤ 0 which contradicts our assumption. (ii) On the contrary, suppose ϕ(0) = t, for some t > 0. Since 0 < t and ϕ is increasing, ϕ(0) < ϕ(t). It follows that t < ϕ(t), which contradicts (i). This concludes the proof. Matkowski function differs from Boyd-Wong functions Defined in Chapter 2. To substantiate this fact, consider the following two functions defined form [0, ∞) to [0, ∞):

ϕ1 ( t ) =

ϕ2 ( t ) =

    0,   

if t = 0,

1 1 1 n+1 , if t ∈ ( n+1 , n ], n = 1, 2, 3, ...,      1, if t > 1,    t , if t < 2, 5

  1 , if t ≥ 2. t

Then ϕ1 is Matkowski but not Boyd-Wong function as it is not upper semi continuous from the right (see [115]). The same function is also used in the form of Example 3 in [87] and Example 2 in [88]. On the other hand, the non-increasing function ϕ2 is Boyd-Wong function but not a Matkowski function. It is worth ´ c [56] proved that for a Boyd-Wong function ϕ, mentioning here that Ciri´ lim ϕn (t) = 0, ∀t > 0.

n→∞

Unfortunately, this is not true in general as pointed out in Jachymski [87]. Matkowski employed his control function to generalized Banach contraction principle as follows:


35

Theorem 3.1.1. Let ( M, d) be a complete metric space and S : M → M. If there exists a Matkowski function ϕ such that d(Su, Sv) ≤ ϕ(d(u, v)), ∀u, v ∈ M, then S has a unique fixed point. First generalization of Theorem 3.1.1 in in a metric space endowed with a partial order was due to Agarwal et al. [8] wherein the authors presented some fixed point results for monotone operators using a weak generalized contraction-type assumption. Definition 3.1.1. Let ( M, d, ) be an ordered metric space. A mapping S : M → M is said to be a generalized Matkowski contraction if there exists a Matkowski function ϕ such that d(Su, Sv) ≤ ϕ( MS (u, v)), ∀u, v ∈ M with u v.

(3.1)

Theorem 3.1.2. [8, Theorem 2.2] Let ( M, d, ) be an ordered metric space and S : M → M. If the following conditions hold: (i) ( M, d, ) is complete, (ii) S is increasing, (iii) there exists u0 ∈ M such that u0 Su0 , (iv) either S is continuous or ( M, d, ) enjoys ICU property, (v) S is a generalized Matkowski contraction. Then S has a fixed point. Moreover, authors in [8] remarked that: Remark 3.1.1. [8, Remark 2.1] The condition (ii) contained in Theorem 3.1.2 can be replaced by "S is decreasing" provided condition (iii) is replaced by "there exists u0 ∈ M such that Su0 u0 ". ´ c et al. [58] utilized the idea of T-monotone It is worth mentioning here that Ciri´ mapping to extend some results of Agarwal et al. [8] to a pair of self-mappings. Moreover, Remark 3.1.1 occurred again but in the setting of a pair of self-mappings in Remark 2.3 of [58].

36


3.2 3.2.1

Fixed point Results Existence Results

The following result remains a sharper version of Theorem 3.1.2: Theorem 3.2.1. Let ( M, d, ) be an ordered metric space, E a subspace of M and S : M → M an increasing mapping such that S( M) ⊆ E. Moreover, suppose that the following conditions hold: (i) E is an O-complete subspace of M, (ii) S is an increasing mapping, (iii) there exists u0 ∈ M such that u0 Su0 , (iv) either S is O-continuous or ( E, d, ) enjoys ICC property, (v) S is a generalized Matkowski contraction. Then S has a fixed point. Proof. Consider u0 ∈ M such that u0 Su0 and define a sequence {un } in E by un := Sn u0 , ∀n ∈ N0 . As S is increasing, we have u0 u1 u2 u3 ... un .... Let dn := d(un+1 , un ). Observe that if dn = 0 for some n ∈ N0 , then un is a fixed point of S and we are through. We assert that: d n ≤ ϕ n ( d0 ). To substantiate the assertion, on setting u = un and v = un−1 in (3.1), we have dn ≤ ϕ( MS (un , un−1 )), where

d ( u n , u n + d ( u n −1 , u n +1 ) MS (un , un−1 ) = max dn−1 , dn , dn−1 , 2 d ( u n −1 , u n +1 ) = max dn−1 , dn , . 2 On using the triangle inequality,

d(un−1 ,un+1 ) 2

≤

d n −1 + d n 2

≤ max {dn−1 , dn }. If

MS (un , un−1 ) = dn , then using part (i) of Proposition 3.1.1, one gets dn ≤ ϕ(dn )
1 equipped with usual metric. Then

( M, d, ≤) is an ordered metric space. Define a self-mapping S on M by Su = u/2 and a Matkowski function ϕ : [0, ∞) → [0, ∞) defined by   1   for t > 1,  2 t,   1 1 1 ϕ(t) = n+1 , for n+1 < t ≤ n ,      0, for t = 0. Then   1   ( u − v ),

2 1 d(Su, Sv) = |u − v| = 1  2   2 ( v − u ),

for u ≥ v, for u ≤ v.

To compute the right hand side of the inequality (3.1), we have d(u, Sv) + d(v, Su) M(u, v) = max d(u, v), d(u, Su), d(v, Sv), 2 1 1 |2u − v| + |2v − u| = max |u − v|, |u|, |v|, 2 2 4    u − v, for − m < v ≤ 2u,       −v/2, for 2u ≤ v ≤ u, =    −u/2, for u ≤ v ≤ 12 u,      v − u, for 1 u ≤ v ≤ 0. 2 We distinguish the following three cases: Case I: If M (u, v) = 0., then the inequality (3.1) holds trivially as d(Su, Sv) = 0. Case II: If 0 < M(u, v) ≤ 1. • Firstly, if v ≤ 2u, then we have 12 (u − v) ≤ ϕ(u − v) = Otherwise, 12 (u − v) >

1 n +1

so that u − v >

2 n +1

1 1 n+1 , for n+1

≥ n1 , a contradiction.

• Secondly, if 2u ≤ v ≤ u, then we have 12 (u − v) ≤ ϕ(−v/2) =

− 12 v ≤ n1 . Otherwise, 21 (u − v) > −v >

1 n +1

< u − v ≤ n1 .

so that

2 1 − u ≥ − u. n+1 n

1 n +1 ,

for

1 n +1

1 2 ≥ . n+1 n

On combining last two inequalities, we get −v/2 > n1 , a contradiction. • Thirdly, if u ≤ v ≤ 12 u, then we have 12 (v − u) ≤ ϕ(−u/2) =

1 n +1 ,

for

1 n +1

1 n +1 ,

that is, v − u >

2 n +1

1 1 n+1 , for n+1

< v − u ≤ n1 .

≥ n1 , a contradiction.

Case III: If M(u, v) > 1 and either v ≤ 2u or u ≤ 2v, then the inequality (3.1) holds trivially. Next, if M(u, v) > 1 and 2u ≤ v ≤ u, then 12 (u − v) ≤ ϕ( M(u, v)) = −v/4. Otherwise, 12 (u − v) > −v/4 implying 2u > y which is a contradiction . With similar arguments, we can show that 12 (u − v) ≤ −u/4 for u ≤ v ≤ 21 u. Therefore, in all cases, the inequality (3.1) holds and S has a fixed point u = 0 supporting Theorem 3.2.1. Remark 3.2.1. since ϕ utilized in Example 2.2.1 is of Matkowski type, Example 2.2.1 can be covered by Theorem 3.2.1 while Theorem 2.1 (of [8]) can not. Remark 3.2.2. Theorem 3.2.1 remains a generalization of Theorems 1.3.1, 1.3.2 and 3.1.2 ´ c et al. [58, Corollary 2.7], O’Regan et al. [130, Theorem 3.6]. besides Ciri´ Now, we furnish an example which serves as a counter example to Remark 3.1.1. Example 3.2.2. Consider M = [1, ∞) equipped with usual metric. Then ( M, d, ) is an ordered metric space wherein the partial order ’’ is defined for u, v ∈ M by: u v ⇔ u ≤ v for u, v ∈ (1, ∞) and 1 1 where ’≤’ is the usual order on R. Then

( M, d, ) is a complete ordered metric space. Define a self-mapping S on M by:   2 if u = 1, Su =  1 otherwise. Define a function ϕ : [0, ∞) → [0, ∞) by    t , for 0 ≤ t ≤ 2, 2(1+ t ) ϕ(t) =  t, for 2 < t. 6 Observe that, ϕ is a Matkowski function. Moreover,

40

C HAPTER 3: R ESULTS FOR M ATKOWSKI T YPE C ONTRACTIONS • u Su, ∀u 6= 1 in M, • S is monotone, • ( M, d, ) enjoys ICU property and • d(Su, Sv) = 0 for all comparable elements u, v ∈ M,

but S is a fixed point free. Remark 3.2.3. With a view to consider Example 3.2.2 for a pair of self-mappings (S, T ) ´ c et al. [58] is on M, if we take T = I M , then one can notice that Remark 2.3 due to Ciri´ also incorrect. We close this section by providing results for if the mapping S in Theorem 3.2.1 is supposed to be decreasing. Observe that, the following result remains a generalization of Theorem 4 contained in Nieto and Rodríguez-López [128]. Theorem 3.2.2. Let ( M, d, ) be an ordered metric space, E a subspace of M and S : M → M be such that S( M ) ⊆ E. Suppose that the following conditions hold: (i) E is an O-complete, (ii) S is a deceasing mapping, (iii) there exists u0 ∈ M such that u0 ≺ Su0 , (iv) either S is an O-continuous or ( E, d, ) enjoys TCC property, (v) S is a generalized Matkowski contraction, (vi) ( M, ) is sequentially chainable. Then S has a fixed point. The proof of above result is very similar to that one of Theorem 2.2.2 and hence omitted.

3.2.2

Uniqueness Results

In a similar manner of Section 2.3, in this subsection, we consider a relatively stronger contraction condition to prove the uniqueness which is contained in the following definition.


41

Definition 3.2.1. Let ( M, d, ) be an ordered metric space. Then S : M → M is said to be a lean generalized Matkowski contraction if there exists a Matkowski function function φ such that d(Su, Sv) ≤ ϕ(mS (u, v)), ∀ u, v ∈ M with u v.

(3.2)

Theorem 3.2.3. Let ( M, d, ) be an ordered metric space, E a subspace of M and S : M → M such that S( M) ⊆ E. Suppose that the following conditions hold: (i) E is O-complete subspace of M., (ii) S is an increasing mapping, (iii) there exists u0 ∈ M such that u0 Su0 , (iv) either S is O-continuous or ( E, d, ) enjoys ICC property, (v) S is lean generalized Matkowski contraction, Then S has a fixed point. Further, if ( M, ) is directed, then S has a unique fixed point. Proof. As every lean generalized Matkowski contraction mapping is a generalized Matkowski contraction, the existence of the fixed point of S follows from Theorem 3.2.1. To prove the uniqueness of such fixed point, let u, v ∈ M be two fixed points of S. We need to show that u = v. By the directedness of M, there exists some w ∈ M such that u ≺ w ≺ v. Set w0 = w and define a sequence {wn } in M by: wn := Sn w0 , ∀n ∈ N. As S is increasing, we have u ≺ wn , ∀ n ∈ N. We claim that lim d(u, wn ) = 0.

n→∞

If d(u, wm ) = 0 for some m ∈ N, then we must have d(Su, wm+1 ) = 0, that is, d(u, wm+1 ) = 0. Consequently, d(u, wn ) = 0, ∀n ≥ m. Otherwise, setting u = u, v = wn in (3.2) gives rise d(u, wn+1 ) = d(Su, Swn )

≤ ϕ(mS (u, wn )),

(3.3)

42


where

d(u, Su) + d(wn , wn+1 ) d(u, wn+1 ) + d(wn , Su) , mS (u, wn ) = max d(u, wn ), 2 2

Obviously, d(u, wn+1 ) + d(wn , u) d ( w n , w n +1 ) ≤ . 2 2 Thus, if

d(u,Swn+1 )+d(wn ,Su) 2

≤ d(u, wn ) then from (3.3), we have d(u, wn+1 ) ≤ ϕ(d(u, wn )).

As ϕ is a Matkowski function, by induction, we get d(u, wn+1 ) ≤ ϕ(d(u, wn )) ≤ ϕ2 (d(u, wn−1 )) ≤ ... ≤ ϕn+1 (d(u, w0 )), which on letting n → ∞ established the claim in this case. The rest of the proof run on the same lines of the proof of Theorem 2.3.1. Remark 3.2.4. One can obtain dual type results corresponding to Theorems 3.2.1 and 3.2.3 if involved terms namely: O-completeness, O-continuity and ICC property are replaced by O-completeness, O-continuity and the DCC property, respectively provided assumption (iii) is replaced by the following: (iiia) there exists u0 ∈ M such that u0 Su0 . Remark 3.2.5. One can obtain similar results corresponding to Theorems 3.2.1 and 3.2.3 if involved terms namely: O-completeness, O-continuity and the ICC property are respectively, replaced by O-completeness, O-continuity and the MCC property, respectively provided assumption (iii) is replaced by the following: (iiib) there exists u0 ∈ M such that u0 ≺ Su0 . Remark 3.2.6. The condition "( M, ) is directed" ensures the uniqueness of the fixed point in Theorem 3.2.2 provided condition (3.1) is replaced by the relatively stronger condition (3.2).

3.2.3

An Application

As an application of Theorem 3.2.1, we prove the following result for a mapping satisfying Matkowski integral type contraction inequality in ordered metric space.


43

To Rdescribe the same, let Π be the set of all functions ω ∈ Λ satisfying u R R0: 0 ω(t)dt ω (t)dt ω (t)dt tends to zero when the number of integral iterates tends 0 to infinity. Theorem 3.2.4. Let ( M, d, ) be an ordered metric space, E an O-complete subspace of M and S : M → M an increasing mapping such that S( M) ⊆ E. Suppose that for every u, v ∈ M with u v and ω ∈ Π, we have d(Su, Sv) ≤

Z ϕ( M(u,v)) 0

ω (t)dt,

(3.4)

where ϕ is a Matkowski function. If assumptions (iii), (iv) and (v) of Theorem 3.2.1 are satisfied, then S has a fixed point. Proof. Define Γ : [0, ∞) → [0, ∞) as in the proof of Theorem 2.4.1. Then (3.4) can be written as d(Su, Sv) ≤ (Γ ◦ ϕ)( M (u, v)). Owing to Theorem 3.2.1, it is enough to show that Γ ◦ ϕ is a Matkowski function. It is obvious that Γ ◦ ϕ is an increasing mapping. Therefore, we show that lim (Γ ◦ n→∞

ϕ)n (t)

= 0. Here, we have R ϕ(t) Rt I1 = (Γ ◦ ϕ)(t) = 0 ω (s)ds ≤ 0 ω (s)ds. R I1 R R0t ω (s)ds R ϕ( I1 ) 2 ω (s)ds. I2 = (Γ ◦ ϕ) (t) = (Γ ◦ ϕ)( I1 ) = 0 ω (s)ds ≤ 0 ω (s)ds ≤ 0 By induction, we get In ≤

Z R: 0

Rt 0 ω (s)ds

ω (s)ds

0

ω (s)ds,

where the integral is taken n times. On letting n → ∞ and using the definition of Π, we get lim In = 0. This completes the proof. n→∞

3.3

Common Fixed point Results

Agarwal et al. in [8] presented the following fixed point theorem for Matkowski type nonlinear contractions in an ordered metric space.

44


Theorem 3.3.1. [8, Theorem 2.1] Let ( M, d, ) be an ordered metric space and S : M → M. Suppose that the following conditions hold: (i) ( M, d) is complete, (ii) S is an increasing mapping, (iii) either S is continuous or ( M, d, ) enjoys ICU property, (iv) there exists u0 ∈ M such that u0 Su0 , (v) there exists a Matkowski function ϕ such that d(Su, Sv) ≤ ϕ(d(u, v)), ∀ u, v ∈ M with u v. Then S has a fixed point. The aim of this section is to extend Theorem 3.3.1 to a pair (S, T ) of self-mappings such that S is T-increasing wherein either M or one of the subspaces S( M ) and T ( M) is complete. Some illustrative examples are also furnished

3.3.1

Existence Results

Theorem 3.3.2. Let ( M, d, ) be an ordered metric space and (S, T ) a pair of self-mappings on M. Suppose that the following conditions hold: (i) S( M) ⊆ T ( M), (ii) S is T-increasing, (iii) there exists u0 ∈ M such that Tu0 Su0 , (iv) there exists a Matkowski function ϕ such that d(Su, Sv) ≤ ϕ(d( Tu, Tv)), ∀ u, v ∈ M with Tu ≺ Tv, (v) (v1) ( M, d) is O-complete, (v2) (S, T ) is O-compatible pair, (v3) T is O-continuous, (v4) (`v) either S is O-continuous or ( M, d, ) has T-ICC property, or alternately (`v) (`v1) either ( T ( M), d, ) or (S( M), d, ) is O-complete, (`v2) either S is (T,O)-continuous or both S and T are continuous or ( T ( M), d, ) enjoys ICC property. Then the pair (S, T ) has a coincidence point.


45

Proof. The proof of this result is divided into two parts. The first part is levelled as Step 1 wherein we use conditions (i), (ii), (iii) and (iv) while the second part is leveled as Step 2 wherein we use conditions embodied in (v) or alternatively, Step 20 where we use conditions contained in (`v). Step 1: Let u0 ∈ M such that Tu0 Su0 and consider the sequence {un } ⊆ M defined by Sun = Tun+1 , ∀ n ∈ N0

(3.5)

Since Tu0 Su0 and Su0 = Tu1 , we have Tu0 Tu1 . Since S is T-increasing we have Su0 Su1 . Continuing this process inductively, we define an increasing sequences { Tun } and {Sun } satisfying (3.5). Notice that if Sun+1 = Sun for any n ∈ N0 , then by (3.5), Sun+1 = Sun = Tun+1 , so that un+1 is a coincidence point of S and T and no need more. So, we may assume such equality does not occur for all n ∈ N0 . Now we show that { Tun } as well as {Sun } are Cauchy sequences. Since the terms of the increasing sequence { Tun } are comparable, then by assumption (iv) we have, d(Su2 , Su1 ) ≤ ϕ(d( Tu2 , Tu1 ))

= ϕ(d(Su1 , Su0 )). By induction, d(Sun+1 , Sun ) ≤ ϕn (d(Su1 , Su0 )) → 0 as n → ∞. Let ε be fixed. Choose n ∈ N0 so that d(Sun+1 , Sun ) < ε − ϕ(ε). Now, d(Sun+2 , Sun ) ≤ d(Sun+2 , Sun+1 ) + d(Sun+1 , Sun )

< ϕ(d( Tun+2 , Tun+1 ) + ε − ϕ(ε) = ϕ(d(Sun+1 , Sun ) + ε − ϕ(ε) < ϕ(ε − ϕ(ε)) + ε − ϕ(ε) < ϕ(ε) + ε − ϕ(ε) = ε.

46


Also, d(Sun+3 , Sun ) ≤ d(Sun+3 , Sun+1 ) + d(Sun+1 , Sun )

< ϕ(d( Tun+3 , Tun+1 )) + ε − ϕ(ε) = ϕ(d(Sun+2 , Sun )) + ε − ϕ(ε) < ϕ(ε) + ε − ϕ(ε) = ε. By induction d(Sun+k , Sun ) < ε, ∀ k ∈ N. Thus, {Sun } is a Cauchy sequence and henceforth by (3.5) so is { Tun }. Step 2: Owing to O-completeness of M and (3.5), there exists some z ∈ M such that Tun ↑ z and Sun ↑ z.

(3.6)

In view of the O-continuity of T, we have lim TTun = lim TSun = Tz.

n→∞

n→∞

(3.7)

Using the O-compatibility of the pair (S, T ), we have lim d( TSun , STun ) = 0.

n→∞

(3.8)

Now, we show that z is a coincidence point of the pair (S, T ) using assumption (v4). To accomplish this, let S be O-continuous. Then using (3.6), we have lim STun = S lim Tun = Sz.

n→∞

n→∞

(3.9)

By combining (3.7), (3.8) and (3.9), we conclude that Sz = Tz and hence we are through. Alternately, let ( M, d, ) has the T-ICC property. Since Tun ↑ z, there exists a subsequence { Tunk } of { Tun } such that TTunk ≺ Tz ∀ k ∈ N0 .

(3.10)

By applying assumption (iv) on (3.10) and using Proposition 3.1.1 (distinguishing


47

the cases whether d( TTunk , Tz) is zero or non-zero), we obtain d(STunk , Sz) ≤ ϕ(d( TTunk , Tz)),

≤ d( TTunk , Tz).

(3.11)

By using (3.7), (3.8), (3.11) and the triangular inequality, we get d( Tz, Sz) ≤ d( Tz, TSunk ) + d( TSunk , STunk ) + d(STunk , Sz)

≤ d( Tz, TSunk ) + d( TSunk , STunk ) + d( TTunk , Tz) → 0 as n → ∞. Thus z ∈ M is a coincidence point of S and T and hence we are also through with this Step as well. step20 : In step 1, we constructed an increasing Cauchy sequences { Tun } and

{Sun }. Let assumption (v1) ` be hold. If ( T ( M ), d, ) is O-complete, then there exists u ∈ M such that Tun ↑ Tu. Alternately, if (S( M ), d ) is O-complete then there exists some w ∈ M such that Sun ↑ Sw. By assumption (i), there exists u ∈ M with Sw = Tu which along with (3.5) give, Tun ↑ Tu, u ∈ M.

(3.12)

Now, by (v2) ` we show that u is a coincidence point of S and T. Firstly, suppose that S is ( T, O)-continuous. From (3.12) we get, Sun ↑ Su, that is, Tun+1 ↑ Su. Now, the uniqueness of the limit implies Tu = Su and hence we are through. Next, suppose that S and T are continuous, then our proof can be completed on the lines of the proof of Theorem 1 in [14] wherein Lemma 1.3.3 is used. Finally, suppose that ( T ( M), d, ) enjoys ICC property. Since Tun ↑ Tu there exists a subsequence { Tunk } of { Tun } such that Tunk ≺ Tu ∀ k ∈ N0 .

48


Using assumption (iv), (3.5) and Proposition 3.1.1 we obtain d( Tunk +1 , Su) = d(Sunk , Su)

≤ ϕ(d( Tunk , Tu)) ≤ d( Tunk , Tu).

(3.13)

On using (3.12), (3.13) and continuity of d, we get d( Tu, Su) = d( lim Tunk +1 , Su) n→∞

= ≤

lim d( Tunk +1 , Su)

n→∞

lim d( Tunk , Tu)

n→∞

= 0 so that Tu = Su. Hence u ∈ M is a coincidence point of the pair (S, T ). Corresponding to Theorem 3.3.2, we can prove a result involving the analogues terms namely: O-completeness, O-compatibility pair, O-continuity, DCC property and T − DCC property as follows. Theorem 3.3.3. Let ( M, d, ) be an ordered metric space and (S, T ) a pair of self-mappings on M. Suppose that the following conditions hold: (i) S( M) ⊆ T ( M), (ii) S is T-increasing, (iii) there exists u0 ∈ M such that Tu0 Su0 , (iv) there exists a Matkowski function ϕ such that d(Su, Sv) ≤ ϕ(d( Tu, Tv)), ∀ u, v ∈ M with Tu ≺ Tv, (v) (v1) ( M, d) is O-complete, (v2) (S, T ) is O-compatible pair, (v3) T is O-continuous, (v4) either S is O-continuous or ( M, d, ) has T-DCC property, or alternately (`v) (`v1) either ( T ( M), d, ) or (S( M), d, ) is O-complete, (`v2) either S is (T,O)-continuous or S and T are continuous or ( T ( M), d, ) has DCC property. Then the pair (S, T ) has a coincidence point.


49

Proof. The proof is analogous to that of Theorem 3.3.2. With a view to avoid any repetition, the proof is omitted. The following example demonstrates the fact that Theorems 3.3.2 and 3.3.3 are genuinely different. Example 3.3.1. Consider M = [0, 1) with the usual metric and usual order. Then ( M, d, ≤

) is an ordered metric space. Define S, T : M → M by Su =

u2 u2 and Tu = ∀ x ∈ M. 4 2

Define ϕ : [0, ∞) → [0, ∞) by ϕ(t) = λt, ∀ t ∈ [0, ∞), where λ ∈ [ 12 , 1). Observe that, ( M, d, ≤) is O-complete and rest of the conditions of the Theorem 3.3.3 are satisfied so that Theorem 3.3.3 is applicable ensuring the existence of the coincidence point (namely u = 0). On the other hand, Cauchy sequence un = 1 −

1 n

does

not converge in M. Also, neither (S( M), d) nor ( T ( M ), d) are complete spaces so that Theorem 3.3.3 can not be used in the context of this example. Combining Theorems 3.3.2 and 3.3.3, we obtain the following. Theorem 3.3.4. Theorem 3.3.2 remains true if involved O-notions and T-ICC property are, respectively, replaced by O-notions and T-MCC property provided assumption (iii) is replaced by the following: ˝ there exists u0 ∈ M such that u0 ≺ Su0 . (iii) Owing to remark 1.3.3, we have the following. Corollary 3.3.1. Theorems 3.3.2–3.3.4 remain true if we replace condition (v2) by one of the following conditions (besides retaining the rest of the hypotheses): (a) (S, T ) is commuting pair, (b) (S, T ) is weakly commuting pair, (c) (S, T ) is compatible pair. On setting T = I M in Theorem 3.3.2, (same can be done for Theorems 3.3.3 and 3.3.4 ), we deduce the following result.

50


Corollary 3.3.2. Let ( M, d, ) be an ordered metric space and S : M → M. Suppose that the following conditions hold: (i) either (S( M), d) or ( M, d) is complete, (ii) S is increasing, (iii) either S is continuous or ( M, d, ) enjoy ICU property, (iv) there exists u0 ∈ M such that u0 Su0 , (v) there exists a Matkowski function ϕ such that d(Su, Sv) ≤ ϕ(d(u, v)), ∀ u, v ∈ M with u ≺ v. Then S has a fixed point. Remark 3.3.1. Above corollary covers Theorem 3.1.2, that is, the completeness of S( M ) is enough to get the result where M may or may not be complete. On setting ϕ(t) = λt (with λ ∈ [0, 1)) and T = I M in Theorem 3.3.2, we deduce the following. Corollary 3.3.3. Corollary 3.3.2 remain true if the condition (v) is replaced by the following condition (besides retaining the rest of the hypotheses): ´(v) there exists λ ∈ [0, 1) such that d(Su, Sv) ≤ λd(u, v), ∀ u, v ∈ M with u ≺ v. Remark 3.3.2. Corollary 3.3.3 (and a fortiori to Corollary 3.3.2 and Theorem 3.3.2 ) covers Theorems 1.3.1, 1.3.2, 1.3.3 and 3.1.2. Observe that, in mentioned theorems the completeness of M can be alternately replaced by the completeness of S( M).

3.3.2

Uniqueness Results

Employing Definition 1.3.3, we prove the following uniqueness result. Theorem 3.3.5. In addition to the hypotheses (i)-(iv) along with (`v) in any one of the Theorems 3.3.2-3.3.4, if one of the following conditions holds, then the pair (S, T ) has a unique point of coincidence:


51

(Q1 ) C (Su, Sv, ≺, T ( M )) is non-empty, for every u, v ∈ M, (Q2 ) ( M, ) is (S, T )-directed, (Q3 ) (S( M), ≺) is totally ordered, Proof. We opt to prove this result corresponding to Theorem 3.3.2 when the condition (Q1 ) holds. The proof of results corresponding to other theorems is similar, hence it is omitted once we show that each of the other conditions implies

( Q1 ). So assume that (Q1 ) holds, i.e., there exist u, v ∈ M such that Tu = Su = u and Tv = Sv = v,

(3.14)

for some u, v ∈ M. We show that u = v. Now, the proof of our theorem is similar to that of Theorem 4 of [15] except proving the claim (15), that is, proving that lim tin = 0 for every 1 ≤ i ≤ k − 1. n→∞

On fixing i, two cases arise: If tim = d( Twim , Twim+1 ) = 0, for some m ∈ N0 , then the proof can be completed on the lines of Theorem 4 of [15]. Secondly, if tim > 0, f or all n ∈ N0 . Then we have tin+1 = d( Twin+1 , Twin++11 )

= d(Swin , Swin+1 ) ≤ ϕ(d( Twin , Twin+1 )) = ϕ(tin ) ≤ ϕ2 (tin−1 ) ≤ ... ≤ ϕn (t1i ) which on making n → ∞ on both the sides gives rise lim tin = 0. Thus, in all, our n→∞

claim is established. In order to show that (Q2 ) ⇒ ( Q1 ), suppose ( Q2 ) holds. Then for every u, v ∈ M,

∃ w ∈ M such that Su ≺ Tw ≺ Sv, which together with condition (i) imply that {Su, Tw, Sv} is a ≺-chain in between Su and Sv. It follows that C(Su, Sv, ≺ , T ( M )) is non-empty for each u, v ∈ M, i.e., ( Q1 ) holds which amounts to say that

( Q2 ) implies ( Q1 ). Similarly, we can show that ( Q3 ) ⇒ ( Q1 ).

52


Theorem 3.3.6. In addition to the hypotheses of Theorem 3.3.5, if the following condition holds, then the pair (S, T ) has a unique coincidence point: (Q4 ) one of S and T is one-one. Proof. Let S be one-one (same argument holds if T is so). Assume there exist two coincidence points u, v ∈ M. By hypothesis there exists a unique point of coincidence which in turn implies that Tu = Su = Tv = Sv. As S is one-one, we have u = v. Theorem 3.3.7. In addition to the hypotheses of Theorem 3.3.5, if the following condition holds, then the pair (S, T ) has a unique common fixed point: (Q5 ) (S, T ) is a weakly compatible pair. Proof. Let u ∈ M be a coincidence point of S and T, then there is u ∈ M such that Tu = Su = u. By Lemma 1.2.1, u itself is a coincidence point, that is Su = Tu. By theorem 3.3.5 we must have u = u, which yields u = Su = Tu, that is, u is a unique common fixed point of the pair (S, T ). Theorem 3.3.8. In addition to the conditions (i)-(v) of the hypotheses of any of the theorems 3.3.2-3.3.4, if any of the conditions ( Q1 ), ( Q2 ) or ( Q3 ) of theorem 3.3.5 holds, then the pair (S, T ) has a unique common fixed point. Proof. By remark 1.3.3, every O-compatibly pair, O-compatibly pair and Ocompatibly pair is a weakly compatible pair. Hence, ( Q5 ) holds trivially. Now, proceeding on the lines similar to the proof of Theorem 3.3.5 our result follows. The following example demonstrates the utility of Theorem 3.3.2 and the corresponding uniqueness theorem, i.e., Theorems 3.3.8. Example 3.3.2. Consider M = R equipped with the usual metric. Then ( M, d, ≤) is an ordered metric space wherein the partial order is defined by: u ≤ v ⇔ |u| ≤ |v| and uv ≥ 0. Define S, T : M → M by Su =

u2 u2 and Tu = , ∀u, v ∈ M. 14 2


53

Clearly S is T-increasing. Define ϕ : [0, ∞) → [0, ∞) by    u if u < 2, 5 ϕ(u) =   u if u ≥ 2. 6 Notice that ϕ is a Matkowski function. Now, ∀u, v ∈ M with Tu ≤ Tv, we have 2 2 2 2 u u v v 2 − = d( Tu, Tv) < ϕ(d( Tu, Tv)). d(Su, Sv) = − = 14 14 14 2 2 7 Observe that, S, T and ϕ satisfy assumptions (i)-(v) of Theorem 3.3.2 so that the pair (S, T ) has a coincidence point in M. Also, observe that condition ( Q2 ) holds, therefore, in view of Theorem 3.3.8, the pair (S, T ) has a unique common fixed point namely: u = 0. Here, it can be pointed out that the corresponding theorems proved in [15] can not be used in the context of above example as ϕ is not a Boyd-Wong function.

Our next example demonstrates the utility of Theorem 3.3.2 and the corresponding uniqueness theorems, i.e., Theorems 3.3.5 and 3.3.6. Example 3.3.3. Consider M = [−8, ∞) equipped with usual metric and usual order. Then

( M, d, ≤) is an ordered metric space. Define S, T : M → M by Su = 2 and Tu = u2 − 7, for all x ∈ M. Also, define ϕ : [0, ∞) → [0, ∞) by ϕ(t) = λt, λ ∈ [0, 1). By a routine calculation one can verify that all the conditions of Theorem 3.3.2 are satisfied except condition (v) (wherein (v2) does not hold). Hence, the pair (S, T ) has a coincidence point in M. Moreover, ( Q1 ) also holds and henceforth, in view of Theorem 3.3.5, the pair

(S, T ) has a unique point of coincidence namely: u = 2. Finally, notice that, ( Q4 ) does not hold, i.e., neither S nor T is one-one, so that Theorem 3.3.6 ensuring the uniqueness of the coincidence point can not be used in the present context. Observe that the involved maps have two coincidence points (namely: u = 3 and u = −3).

CHAPTER

4

Results for Generalized Weak Contractions

Inspired by a metrical-fixed point theorem due to Choudhury et al. [50], in this chapter, we prove some order-theoretic common fixed point results which generalize several core results of the existing literature, especially the two main results of Harjani and Sadarangani [77, 78]. We demonstrate the realized improvements in our results using an illustrative example. As an application of our main result herein, we prove a result for a pair of mappings satisfying an integral type (ψ, ϕ) T -generalized weak contractive condition.

4.1


Recall that a self-mapping S on a metric space ( M, d) is said to be ϕ-weakly contractive if d(Su, Sv) ≤ d(u, v) − ϕ(d(u, v)) for all u, v ∈ M, where ϕ is an altering distance functions (see Definition 1.2.5). In order to extend Banach contraction principle, Alber and Guerre-Delabriere [16] utilized the same to prove some fixed point results on Hilbert spaces. Thereafter, Rhoades [154] proved that the main result contained in [16] remains true for complete metric spaces. The content of this chapter is based on the following article: R. Gubran and M. Imdad Results on Coincidence and Common Fixed Points for (ψ, ϕ) g -Generalized Weakly Contractive Mappings in Ordered Metric Spaces Mathematics, Vol. 4 Art. ID: 68 (2016).

55

56

C HAPTER 4: R ESULTS FOR G ENERALIZED W EAK C ONTRACTIONS

Theorem 4.1.1. [154] Let ( M, d) be a complete metric space. Then every ϕ-weakly contractive mapping S : M → M has a unique fixed point. In this regard, it is worth noting that, Alber and Guerre-Delabriere [16] assumed that the altering distance function ϕ satisfies an extra condition, i.e., lim ϕ(t) = ∞, t→∞

but Rhoades [154] proved above result without this condition. Thereafter, Dutta and Choudhury [66] proved a generalization of Theorem 4.1.1 utilizing the following definition which remains a generalization of Definition 1.2.6: Definition 4.1.1. [66] Let ( M, d) be a metric space. A mapping S : M → M is said to be

(ψ, ϕ)-weakly contractive if ψ(d(Su, Sv)) ≤ ψ(d(u, v)) − ϕ(d(u, v)), ∀u, v ∈ M,

(4.1)

where ψ and ϕ are altering distance functions. Theorem 4.1.2. [66] Let ( M, d) be a complete metric space and S : M → M a (ψ, ϕ)weakly contractive mapping. Then S has a unique fixed point. Choudhury et al.

[50] introduced the following notion and proved a

generalization of the above two theorems as under: Definition 4.1.2. [50] Let ( M, d) be a metric space. A mapping S : M → M is said to be

(ψ, ϕ)-generalized weakly contractive mapping if ψ(d(Su, Sv)) ≤ ψ( MS (u, v)) − ϕ(max {d(u, v), d(v, Sv)}), ∀u, v ∈ M,

(4.2)

where ψ is an altering distance function and ϕ : [0, ∞) → [0, ∞) is a continuous function with ϕ(t) = 0 if and only if t = 0. Theorem 4.1.3. [50] Let ( M, d) be a complete metric space and S : M → M a (ψ, ϕ)generalized weakly contractive mapping. Then S has a unique fixed point. In recent years, the idea of weak contraction has been exploited by several researchers (e.g., [2, 3, 26, 27, 34, 36, 51, 66, 82, 120, 123, 131, 149, 182, 183, 192].) Remark 4.1.1. With a view to emphasize the order-theoretic analogue of Definition 4.1.1 (resp. Definition 4.1.2), it can be pointed out that the inequality (4.1) (resp. (4.2)) is required


57

to hold merely for comparable elements, i.e., for all u, v ∈ M such that u v rather than for every pair of elements in M (see Remark 1.3.1). Harjani and Sadarangani [77] proved an order-theoretic analogue of Theorem 4.1.1 as follows: Theorem 4.1.4. [77, Theorems 2 and 3] Let ( M, d, ) be a complete ordered metric space and S : M → M. Suppose that the following conditions hold: (i) S is increasing, (ii) S is a ϕ-weakly contractive mapping with lim ϕ(t) = ∞, t→∞

(iii) either S is a continuous mapping or ( M, d, ) enjoys ICU property. Then S has a fixed point provided there exists u0 ∈ M such that u0 Su0 . Subsequently, Harjani and Sadarangani [78] proved the following an ordertheoretic analogue of Theorem 4.1.2 as well as a generalization of Theorem 4.1.4. Theorem 4.1.5. [78] (Theorems 2.1 and 2.2) Let ( M, d, ) be a complete ordered metric space and S : M → M . Suppose that the following conditions hold: (i) S is increasing, (ii) S is a (ψ, ϕ)-weakly contractive mapping, (iii) either S is a continuous mapping or ( M, d, ) enjoys ICU property. Then S has a fixed point provided there exists u0 ∈ M such that u0 Su0 . Furthermore, Harjani and Sadarangani [78] assumed that ( M, ) is a directed set as a sufficient condition for the uniqueness of the fixed point in Theorem 4.1.5.

In this chapter, we prove an order-theoretic analogue of Theorem 4.1.3 so as to improve and generalize Theorems 4.1.4 and 4.1.5 in the following aspects: (a) relatively weaker notions of the continuity and completeness are employed, (b) the (ψ, ϕ)-weak contractive condition is replaced by a (ψ, ϕ) T -generalized weak contractive condition (defined later) involving a pair of self-mappings, (c) a weaker uniqueness condition is utilized.

58

C HAPTER 4: R ESULTS FOR G ENERALIZED W EAK C ONTRACTIONS We demonstrate the genuineness of our results by a suitable example. As

an application, we prove a result for mappings satisfying integral type (ψ, ϕ) T generalized weak contractive condition.

4.2

Coincidence Point Results

In the sequel, we use the following definition: Definition 4.2.1. Let ( M, d, ) be an ordered metric space and S, T : M → M. Then S is said to be a (ψ, ϕ) T -generalized weakly contractive mapping if ψ(d(Su, Sv)) ≤ ψ( MS,T (u, v)) − ϕ(max {d( Tu, Tv), d( Tv, Sv)}),

(4.3)

for all u, v ∈ M with Tu Tv where ψ is an altering distance function and ϕ : [0, ∞) → [0, ∞) is a lower-semi continuous function with ϕ(t) = 0 ⇔ t = 0. Observe that, on setting T = I M , Definition 4.2.1 remains relatively weaker than the order-theoretic analogue of Definition 4.1.2 as the class of lower-semi continuous functions is relatively larger than the class of continuous functions.

Now, we prove our main result as follows: Theorem 4.2.1. Let ( M, d, ) be an ordered metric space, E an O-complete subspace of M and S, T : M → M. Suppose the following conditions hold: (i) S is T-increasing, (ii) S is a (ψ, ϕ) T -generalized weakly contractive mapping, (iii) (a) S( M) ⊆ E ⊆ T ( M ), (b) either S is ( T, O)-continuous or S and T are continuous or ( E, d, ) enjoys ICU property, (iv) there exists u0 ∈ M such that Tu0 Su0 . Then the pair (S, T ) has a coincidence point. Proof. Choose u0 ∈ M such that Tu0 Su0 . As the mapping S is T-increasing and S( M) ⊆ T ( M ), we can define increasing sequences { Tun } and {Sun } in M such


59

that, for all n ∈ N0 , Tun+1 = Sun .

(4.4)

Observe that, { Tun } and {Sun } are in E. Moreover, if d( Tun , Tun+1 ) = 0, for some n ∈ N0 , then un is the required coincidence point and we are done. Henceforth, we assume that d( Tun , Tun+1 ) > 0, ∀n ∈ N0 . We assert that lim d( Tun , Tun+1 ) = 0. On setting u = un , v = un+1 in (4.3), we n→∞

get ψ(d( Tun+1 , Tun+2 )) = ψ d(Sun , Sun+1 )

≤ ψ( MS,T (un , un+1 ))− ϕ max d( Tun , Tun+1 ), d( Tun+1 , Tun+2 ),

(4.5)

for all n ∈ N0 , where MS,T (un , un+1 ) = max d( Tun , Tun+1 ), d( Tun , Sun ), d( Tun+1 , Sun+1 ) d( Tun , Sun+1 ) + d( Tun+1 , Sun ) , 2

) d( Tun , Tun+2 ) = max d( Tun , Tun+1 ), d( Tun+1 , Tun+2 ), . 2

By the triangular inequality, max {d( Tun , Tun+1 ), d( Tun+1 , Tun+2 )} ≥

d( Tun ,Tun+2 ) . 2

If possible, assume MS,T (un , un+1 ) = d( Tun+1 , Tun+2 ), then d( Tun , Tun+1 )) ≤ d( Tun+1 , Tun+2 ) so that (4.5) reduces to ψ(d( Tun+1 , Tun+2 )) ≤ ψ(d( Tun+1 , Tun+2 ) − ϕ(d( Tun+1 , Tun+2 ))

< ψ(d( Tun+1 , Tun+2 )), a contradiction. Thus, MS,T (un , un+1 ) = d( Tun , Tun+1 ) and (4.5) becomes ψ(d( Tun+1 , Tun+2 )) ≤ ψ(d( Tun , Tun+1 )) − ϕ(d( Tun , Tun+1 ))

< ψ(d( Tun , Tun+1 )). As ψ is an increasing function, {d( Tun , Tun+1 )} is a decreasing sequence of positive real numbers so that lim d( Tun , Tun+1 ) = r ≥ 0.

n→∞

60


On taking the limit superior as n → ∞ in inequality (4.5), we obtain lim sup ψ(d( Tun+1 , Tun+2 )) ≤ lim sup ψ(d( Tun , Tun+1 ))−

n→∞

n→∞

lim inf ϕ(d( Tun , Tun+1 ))

n→∞

which implies that ψ(r ) ≤ ψ(r ) − ϕ(r ), a contradiction. Therefore, r = 0 so that the assertion established. Now, we claim that { Tun } is a Cauchy sequence in E. For if it is not Cauchy, owing to Lemma 1.3.1, there exist e > 0 and two subsequences { Tunk } and { Tumk } of { Tun } such that nk > mk ≥ k, d( Tumk , Tunk ) ≥ e, d( Tunk −1 , Tumk ) < e and lim d( Tumk , Tunk ) = lim d( Tumk +1 , Tunk )

k→∞

k→∞

= lim d( Tumk , Tunk +1 ) k→∞

= lim d( Tumk +1 , Tunk +1 ) k→∞

= e. Since nk > mk , on putting u = unk and v = umk in (4.3), we have (for all k ∈ N) ψ(d( Tunk +1 , Tumk +1 )) = ψ(d(Sunk , Sumk ))

≤ ψ( MS,T (unk , umk )) − ϕ(max {d( Tunk , Tumk ), d( Tumk , Tumk +1 )}) (4.6) where MS,T (unk , umk )) = max d( Tunk , Tumk ), d(( Tunk , Tunk +1 ), d( Tumk , Tumk +1 ), d( Tunk , Tumk +1 ) + d( Tumk , Tunk +1 ) , . 2 Taking limit superior as n → ∞ in (4.6), we have ψ ( e ) ≤ ψ ( e ) − ϕ ( e ), a contradiction. Thus, { Tun } is a Cauchy sequence in E establishing our claim. Owing to the completeness of E, there exists some u ∈ E such that Tun ↑ u.

(4.7)


61

Due to the condition (iii)a, there exists some w ∈ M such that u = Tw so that Tun ↑ Tw.

(4.8)

Now, using the condition (iii)b, we show that w is a coincidence point of the pair

(S, T ). Firstly, assume that S is ( T, O)-continuous. In view of (4.8), we have Sun → Sw which (in view of (4.4)) by the uniqueness of the limit implies Tw = Sw. Secondly, let S and T be continuous mappings. Then the proof can be outlined on the lines of the proof of Theorem 1 in [15]. Lastly, assume that ( E, d, ) enjoys ICU property. Then Tun Tw, for all n ∈ N0 . On setting u = un , v = w in (4.3), we have ψ(d( Tun+1 , Sw)) = ψ(d(Sun , Sw))

≤ ψ( MS,T (un , w)) − ϕ(max {d( Tun , Tw), d( Tw, Sw)}),

(4.9)

where MS,T (un , w) = max d( Tun , Tw), d( Tun , Tun+1 ), d( Tw, Sw), d( Tun , Sw) + d( Tw, Tun+1 ) . 2 On using (4.4), (4.8) and taking limit superior in (4.9) as n → ∞, we have ψ(d( Tw, Sw)) ≤ ψ(d( Tw, Sw)) − ϕ(d( Tw, Sw)), a contradiction unless Tw = Sw. This concludes the proof. Theorem 4.2.2. Theorem 4.2.1 remains true if assumptions embodied in the condition (iii) are replaced by the following (besides retaining the rest of the hypotheses). ˜ (iii) (a) S( M ) ⊆ E ∩ T ( M ), (b) T is O-continuous, (c) (S, T ) is O-compatible pair and (d) either S is O-continuous or ( E, d, ) enjoys T-ICU property. Proof. The proof runs on the lines of the proof of Theorem 4.2.1 except wherever we used conditions in (iii), which can be altered as follows: Owing to (4.4) and (4.7), we have Sun ↑ u and Tun ↑ u,

(4.10)

62


where u ∈ E. In view of the condition (˜iii)b, we have lim T (Sun ) = Tu = lim T ( Tun ).

n→∞

n→∞

Also, in view of the condition (˜iii)c, we have lim d( T (Sun ), S( Tun )) = 0 so that n→∞

lim S( Tun ) = Tu.

n→∞

Now, on using the condition (˜iii)d, we show that u is a coincidence point of S and T. Let S be O-continuous. Then from (4.10), we have lim S( Tun ) = S( lim Tun ) = Su.

n→∞

n→∞

Combining last two equations, we get Su = Tu and hence we are done. Alternately, let ( E, d, ) enjoy T-ICU property. By (4.10), we have T ( Tun ) Tu for all n ∈ N0 . On putting u = Tun , v = u in (4.3), we get ψ(d(STun , Su)) ≤ ψ( MS,T ( Tun , x )) − ϕ(max {d( TTun , Tu), d( Tu, Sv)),

(4.11)

for all n ∈ N0 , where, MS,T ( Tun , x )) = max d( TTun , Tu), d( TTun , STun ), d( Tu, Su), d( TTun , Su) + d( Tu, STun ) . 2 On taking the limit of (4.11) as n → ∞, we arrive at a contradiction unless Tu = Su. This concludes the proof. ˜ Remark 4.2.1. Observe that the condition (iii)a utilized in Theorem 4.2.2 is relatively weaker than the condition (iii)a of Theorem 4.2.1. On setting T = I M in Theorems 4.2.1 and 4.2.2, we deduce the following: Corollary 4.2.1. Let ( M, d, ) be an ordered metric space, E an O-complete subspace of M and S : M → M such that S( M) ⊆ E. Suppose the following conditions hold: (i) S is increasing, (ii) S is a (ψ, ϕ)-generalized weakly contractive mapping, (iii) either S is O-continuous or ( E, d, ) enjoys ICU property, (iv) there exists u0 ∈ M such that Tu0 Su0 . Then S has a fixed point.


63

Remark 4.2.2. If MS (u, v) = d(u, v), then Corollary 4.2.1 reduces to a sharpened version of Theorem 4.1.5, as the increasing condition on the altering distance function ϕ is found unnecessary and a weaker notion of the continuity of ϕ is utilized. Remark 4.2.3. If MS (u, v) = d(u, v) and ψ := I[0,∞)] in Corollary 4.2.1, we get Theorem 4.1.4 without the assumption lim ϕ(t) = ∞. t→∞

Remark 4.2.4. The completeness in Theorems 4.1.4 and 4.1.5 is merely required on any subspace rather than the whole space M such that this subspace contains S( M). Further, these results can be obtained utilizing a relatively weaker notion of the continuity and completeness. Example 4.2.1. Consider M = (−1, 0] endowed with the usual metric d. Then ( M, d, ) is an O-complete ordered metric space wherein the partial order ‘’ is defined by: u v if and only if u ≤ v for u, v ∈ (−1, 0) and 0 0. Define ψ, ϕ : [0, ∞) → [0, ∞) by ψ(t) = 3t and ϕ := I[0,∞) . Consider S and T two self-mappings on M defined by: S(u) =

1 2 u and T (u) = u. 3 3

Then the left hand side of the inequality (4.3) is ψ(d(Su, Sv)) = |u − v| =

  u − v,

for v ≤ u,

 v − u,

for v ≥ u.

To compute the right hand side of the inequality, we have n d( Tu, Sv) + d( Tv, Su) o ψ ( MS,T (u, v)) = ψ max d( Tu, Tv), d( Tu, Su), d( Tv, Sv), 2 n2 o 1 1 1 = ψ max |u − v|, |u|, |v|, (|u − 2v| + |v − 2u|) 3 3 3 6   2(u − v), for − 1 < v ≤ 2u,       −v, for 2u ≤ v ≤ u, =    −u, for u ≤ v ≤ 21 u,      2(v − u), for 1 u ≤ v ≤ 0 2

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C HAPTER 4: R ESULTS FOR G ENERALIZED W EAK C ONTRACTIONS and

1 2 | u − v |, | v | 3 3

ϕ (max {d( Tu, Tv), d( Tv, Sv)}) = max   2   (u − v), for − 1 < v ≤ 2u,   3 = − 13 v, for 2u ≤ v ≤ 32 u,       2 (v − u), for 2 u ≤ v ≤ 0. 3 3 Thus, the right hand side of (4.3) is

ψ( MS,T (u, v)) − ϕ(max {d( Tu, Tv), d( Tv, Sv)}) =

  4    3 ( u − v ),       − 23 v,   

−u + 31 v,       − 13 (u + 2v),        4 ( v − u ), 3

for − 1 < v ≤ 2u, for 2u ≤ v ≤ u, for u ≤ v ≤ 23 u, for 23 u ≤ v ≤ 12 u, for

u 2

< v ≤ 0.

By a routine calculation, we can see that inequality (4.3) is satisfied, that is, S is a (ψ, ϕ) T generalized weakly contractive mapping and the pair (S, T ) has a coincidence point (namely u = 0) supporting Theorems 4.2.1 and 4.2.2. Observe that, on setting T := I M in Example 4.2.1, we create a situation wherein neither Theorem 4.1.4 nor Theorem 4.1.5 can be used, as the whole space is not complete while our Corollary 4.2.1 works. This substantiates the genuineness of our results proved in this paper. Definition 4.2.2. Let ( M, d, ) be an ordered metric space and (S, T ) a pair of selfmappings on M. Then S is said to be a lean (ψ, ϕ) T -generalized weakly contractive mapping if ψ(d(Su, Sv)) ≤ ψ(mS,T (u, v)) − ϕ(max {d( Tu, Tv), d( Tv, Sv)}),

(4.12)

for all u, v ∈ M such that Tu Tv, where ψ is an altering distance function and ϕ : [0, ∞) → [0, ∞) is a continuous function with ϕ(t) = 0 if and only if t = 0. As mS,T (u, v) ≤ MS,T (u, v), Definition 4.2.1 is weaker than Definition 4.2.2.


65

Corollary 4.2.2. Theorem 4.2.1 remains true if the condition (ii) is replaced by the following condition besides retaining the rest of the hypothesis. ˝ S is a lean (ψ, ϕ) T -generalized weakly contractive mapping. (i) Corollary 4.2.3. Theorem 4.2.2 remains true if the condition (ii) is replaced by the condition ˝ besides retaining the rest of the hypothesis. (i)

4.3

Common Fixed Points Results

Theorem 4.3.1. In addition to the hypotheses of Corollary 4.2.2, if ( M, ) is (S, T )directed, then the pair (S, T ) has a unique point of coincidence. Proof. Let u, v, u, v ∈ M be such that Tu = Su = u and Tv = Sv = v. We assert that u = v. By the hypothesis, there exists w ∈ M such that Tw is comparable to both Su and Sv. For Su ≺ Tw, we may assume Su Tw (other case is similar). Set w0 = w. Since S( M) ⊆ T ( M ) and S is a T-increasing mapping, one can define a sequence {wn } ⊂ M such that Twn+1 = Swn and Tu Twn for all n. We assert that lim d( Tu, Twn ) = 0.

n→∞

(4.13)

To establish the assertion, we distinguish two cases: Firstly, if d( Tu, Twm ) = 0 for some m ∈ N. Then by Lemma 1.3.2, d(Su, Swm ) = 0, that is, d( Tu, Twm+1 ) = 0. On using induction on m, d( Tu, Twn ) = 0 for all n ≥ m establishing the assertion in this case. Secondly, if d( Tu, Twn ) > 0 for all n ∈ N0 , then on setting u = u and v = wn in (4.12), we get ψ(d( Tu, Twn+1 )) = ψ(d(Su, Swn ))

≤ ψ(mS (u, wn )) − ϕ(max {d( Tu, Twn ), d( Twn , Swn )}), for all n ∈ N0 , where

(4.14)

66


mS (u, wn ) = max d( Tu, Twn ),

Obviously,

d( Twn ,Twn+1 ) 2

≤

d( Tu, Twn ). Then mS (u, wn )

d( Tu, Su) + d( Twn , Twn+1 ) , 2 d( Tu, Twn+1 ) + d( Twn , Tu) . 2

d( Tu,Twn+1 )+d( Twn ,Tu) . Assume that d( Tu, Twn+1 ) > 2 d( Twn ,Tu) = d(Tu,Twn+1 )+ . Therefore, from (4.14), we have 2

d( Tu, Twn+1 ) + d( Twn , Tu) . ψ(d( Tu, Twn+1 )) < ψ 2

As ψ is increasing, we have d( Tu, Twn+1 ) ≤ d( Twn , Tu), a contradiction to our assumption. Hence, d( Tu, Twn+1 ) ≤ d( Tu, Twn ) so that mS (u, wn ) = d( Tu, Twn ) and (4.14) reduces to ψ(d( Tu, Twn+1 )) ≤ ψ(d( Tu, Twn )), for all n ∈ N0 . Now, {d( Tu, Twn )} is a decreasing sequence of strictly positive real numbers which must posses a limit r ≥ 0. Letting n → ∞ in (4.14), we get ψ(r ) ≤ ψ(r ) − ϕ(2r ) which is a contradiction unless r = 0. Thus, in all, our assertion is established. Similarly, when Sv ≺ Tw, one can show that lim d( Tv, Twn ) = 0.

n→∞

(4.15)

On using triangular inequality, (4.13) and (4.15), we have d(u, v) = d( Tu, Tv) ≤ d( Tu, Twn ) + d( Twn , Tv) → 0 as n → ∞, which shows that the pair (S, T ) has a unique point of coincidence. Theorem 4.3.2. In addition to the hypotheses of Theorem 4.3.1, if the pair (S, T ) is weakly compatible, then the pair has a unique common fixed point. Proof. Let u ∈ M be an arbitrary coincidence point of the pair (S, T ). Due to Theorem 4.3.1, there exists a unique point of coincidence w ∈ M (say) such that Su = Tu = w. By Lemma 1.2.1, w itself is a coincidence point, i.e., Sw = Tw. Now, again, Theorem 4.3.1 ensures that Sw = Tw = w, i.e., w is a unique common fixed point of S and T. Theorem 4.3.3. In addition to the hypotheses of Corollary 4.2.3, if ( M, ) is (S, T )directed, then the pair (S, T ) has a unique common fixed point.


67

Proof. On the lines of the proof of Theorem 4.3.1, one can show that the pair (S, T ) has a unique point of coincidence. In view of the hypothesis (condition (˜iii)c of Theorem 4.2.2), (S, T ) is an O-compatible pair and hence is a weakly compatible pair (by Remark 1.3.3). Now, the proof can be completed on the lines of the proof of Theorem 4.3.2. Remark 4.3.1. One can obtain dual type results corresponding to all results in Sections 4.2 and 4.3 by replacing “O-analogues” with “O-analogues” and “ ICU property” with “DCL property” provided the existence of u0 ∈ M such that Tu0 Su0 is replaced by the existence of u0 ∈ M such that Tu0 Su0 . Remark 4.3.2. By using Zermelo’s well-ordering Theorem, the set M can be well ordered and the contraction conditions in all above results of Sections 4.2 and 4.3 are valid for each u, v ∈ M. Therefore, each of Theorems 4.3.2 and 4.3.3 covers Theorems 4.1.1, 4.1.2, 4.1.3 as well as Theorem 2.1 of [34]

4.4

An Application

As an application of Theorem 4.2.1 (same can be done for Theorem 4.2.2), we have the following result on coincidence point for mappings satisfying integral type

(ψ, ϕ)T -weakly contraction in ordered metric space. Let Λ be the set of functions ω : [0, ∞) → [0, ∞) satisfying the following: (a) ω is a Lebesgue-integrable mapping on each compact subset of [0, ∞), Re (b) 0 ω (t)dt > 0 for all e > 0. Theorem 4.4.1. Let ( M, d, ) be an ordered metric space and E an O-complete subspace of M. Let (S, T ) be a pair of self-mappings on M such that S is T-increasing. Suppose that for every u, v ∈ M with u v and ω ∈ Λ, we have Z ψ(d(Su,Sv)) 0

ω (t)dt ≤

Z ψ( MS,T (u,v)) 0

ω (t)dt −

Z ϕ(max{d( Tu,Tv),d( Tv,Sv)}) 0

ω (t)dt, (4.16)

68


where ψ and ϕ are as in Definition 4.2.1. If there exists u0 ∈ M such that Tu0 Su0 and the condition (iii) of Theorem 4.2.1 is satisfied, then the pair (S, T ) has a coincidence point. Proof. Define Γ : [0, ∞) → [0, ∞) by Γ(u) =

Z u 0

ω (t)dt,

then (4.16) can be written as Γ ψ d(Su, Sv) ≤ Γ ψ MS,T (u, v) − Γ ϕ max {d( Tu, Tv), d( Tv, Sv)} . Since Γ ◦ ψ : [0, ∞) → [0, ∞) is an altering distance function and Γ ◦ ϕ : [0, ∞) →

[0, ∞) is a lower semi-continuous function with (Γ ◦ ϕ)(t) = 0 if and only if t = 0, the desired result follows from Theorem 4.2.1 .

CHAPTER

Results Under an Implicit Function

5

In this chapter, we prove order-theoretic coincidence and common fixed point theorems for a pair of self-mappings satisfying a unified-type condition governed by an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize many relevant results of the existing literature. Interestingly, unlike several other cases, our main results deduce a nonlinear order-theoretic version of a well-known ´ c [55] proved for quasi-contraction. Moreover, we fixed point theorem due to Ciri´ observe that Theorem 2 contained in Berinde and Vetro [38] is not correct in its present form. Finally, in the setting of metric spaces, we drive a sharpened version of Theorem 1 contained in [38]. As an application, we establish an existence result for a solution of Volterra type integral equation.

5.1

Introduction

Prior to Popa [142, 143], researchers of metric fixed point theory were required to prove a theorem for every contraction condition which amounts to saying that once there is a relatively new contraction, one is required to prove a separate theorem for The content of this chapter is based on an article accepted in J. Adv. Math. Stud. 11 (3) 2018, 481–495.

69

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C HAPTER 5: R ESULTS U NDER AN I MPLICIT F UNCTION

the same. But Popa [142] initiated the idea of implicit function with a view to cover several contraction conditions in one go. In recent years, the idea of implicit function has been utilized by several authors. One of the interesting articles on this theme is due to Berinde and Vetro [38] wherein authors proved results on coincidence as well as common fixed point for a general class of self-mappings covered under an implicit function in the settings of metric and ordered metric spaces. However, we observe that the order-theoretic result of this article is not correct in its present form. Definition 5.1.1. [13] Let ( M, ) be an ordered set. A mapping S : M → M is said to be comparable if it maps comparable elements to comparable elements.

5.2

Implicit function

Recall that a mapping ϕ : [0, ∞) → [0, ∞) is said to be a Matkowski function (see Definition 1.2.4) if ϕ is increasing in [0, ∞) and lim ϕn (t) = 0, for all t > 0. Now, n→∞

we consider the family I of all real continuous functions I : [0, ∞)6 → R. In the respect of the family I , the following conditions will be utilized in our results: (I1a ) I is decreasing in the fifth variable and there exists a Matkowski function ϕ such that I ( x, y, y, x, x + y, 0) ≤ 0, for all x, y ≥ 0, implies that x ≤ ϕ(y). (I1b ) I is decreasing in the fourth variable and there exists a Matkowski function ϕ such that I ( x, y, 0, x + y, x, y) ≤ 0, for all x, y ≥ 0, implies that x ≤ ϕ(y). (I1c ) I is decreasing in the third variable and there exists a Matkowski function ϕ such that I ( x, y, x + y, 0, y, x ) ≤ 0, for all,x, y ≥ 0 implies that x ≤ ϕ(y). (I2 ) I ( x, x, 0, 0, x, x ) > 0, for all x > 0. In [37], Berinde considered the family IB of all real continuous functions I :

[0, ∞)6 → R and the following conditions: (i1a ) I is decreasing in the fifth variable and I ( x, y, y, x, x + y, 0) ≤ 0, for all x, y ≥ 0, implies that there exists h ∈ [0, 1) such that x ≤ hy.


71

(i1b ) I is decreasing in the fourth variable and I ( x, y, 0, x + y, x, y) ≤ 0, for all x, y ≥ 0, implies that there exists h ∈ [0, 1) such that x ≤ hy. (i1c ) I is decreasing in the third variable and I ( x, y, x + y, 0, y, x ) ≤ 0, for all x, y ≥ 0, implies that there exists h ∈ [0, 1) such that x ≤ hy. (i2 ) I ( x, x, 0, 0, x, x ) > 0, for all x > 0. Observe that IB ⊆ I . Definition 5.2.1. If ρ is a Matkowski function such that ϕ defined by: ϕ(t) = ρ(2t), for all t ≥ 0, is a Matkowski function, then ρ is said to be a half-Matkowski function. Observe that the Matkowski function ρ(t) = kt is a half-Matkowski for k ∈ [0, 1/2) while it is not for k ∈ [1/2, 1). Proposition 5.2.1. Let ρ be a half-Matkowski function. Then ρ is a Matkowski function with ρ(2t) < t, for all t > 0. Proof. As ρ is a half-Matkowski function, there exists a Matkowski function ϕ such that ϕ(t) = ρ(2t). Thus, ρ(2t) = ϕ(t) < t, for all t > 0. The following functions satisfy variety of the conditions ( I1a ) − ( I2 ). In all the following examples, ψ is a continuous Matkowski function while ρ is a continuous half-Matkowski function. Example 5.2.1. All functions I defined in Examples 3.1-3.8, 3.17 and 3.19 of [37] are in I and satisfy conditions ( I1a ) − ( I2 ) for ϕ(t) = kt with a suitable k. Example 5.2.2. Define I ∈ I given by: I (t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ρ max {t2 , t3 , t4 , t5 , t6 } . Let x, y ≥ 0 and choose a Matkowski function ϕ where ϕ(t) = ρ(2t), ∀ t ∈ [0, ∞). Assume x ≥ y so that x − ρ( x + y) ≤ 0 ⇒ x ≤ ρ(2x ) = ϕ( x ) which is a contradiction. Thus, x ≤ y and x ≤ ρ( x + y) ≤ ϕ(y). Therefore, I satisfies ( I1a ) with ϕ given by ϕ(t) = ρ(2t), t > 0. Similarly, we can prove that I satisfies ( I1b ), ( I1c ) and ( I2 ) for the same ϕ.

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Example 5.2.3. [38, Example 2] Consider the function I ∈ I , given by: I (t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − k max {t2 , t3 , t4 , t5 , t6 } , k ∈ [0, 1/2). Then I satisfies ( I1a ) − ( I2 ) with ϕ(t) =

kt 1− k , t

≥ 0.

Remark 5.2.1. I defined in Example 5.2.3 is a special case of I defined in Example 5.2.2. Example 5.2.4. Define I ∈ I given by: I ( t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ρ t2 . Then I satisfies ( I1a ) − ( I2 ) with ϕ given by ϕ(t) = ρ(2t), t > 0. Example 5.2.5. Define I ∈ I as:

t5 + t6 I ( t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ψ t3 . t2 + t4 Then I satisfies ( I1a ) and ( I2 ) with ϕ = ψ but does not satisfy ( I1b ) and ( I1c ). Example 5.2.6. Define I ∈ I as:

t5 + t6 I ( t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ψ t2 . t3 + t4 Then I satisfies ( I1a ) and ( I1c ) with ϕ = ψ, while ( I2 ) is not applicable. Example 5.2.7. Define I ∈ I as: I ( t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ψ ( t2 ). Then I satisfies ( I1a ) − ( I2 ) with ϕ = ψ. Example 5.2.8. Define I ∈ I as: I ( t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ρ ( t3 + t4 ). Then I satisfies ( I1a ) − ( I2 ) with ϕ given by ϕ(t) = ρ(2t), t > 0. Observe that, if we replace ρ by ψ in this example, then I satisfies condition ( I2 ) only. Example 5.2.9. Define I ∈ I as: I ( t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ρ ( t2 + t3 ). Then I satisfies ( I1a ), ( I1b ) and ( I2 ) with ϕ given by ϕ(t) = ρ(2t), t > 0. Observe that, if we replace ρ by ψ in Example 5.2.9, then I satisfies conditions ( I1b ) and ( I2 ).


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Example 5.2.10. Define I ∈ I as:

t3 + t4 , t5 , t6 I (t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ψ max t2 , 2

.

Then I satisfies ( I1b ) − ( I2 ) with ϕ = ψ. Example 5.2.11. Define I ∈ I as:

t5 + t6 I (t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ψ max t2 , t3 , t4 , 2

.

Then I satisfies ( I1a ) and ( I2 ) with ϕ = ψ. Example 5.2.12. Define I ∈ I as: t5 + t6 I (t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − ψ max t2 , t3 , t4 , − Lmin{t3 , t4 , t5 , t6 }, L ≥ 0. 2 Then I satisfies ( I1a ) and ( I2 ) with ϕ = ψ. Example 5.2.13. Define I ∈ I as:

1 1 I (t1 , t2 , t3 , t4 , t6 ) = t1 − ψ max t2 , [t3 + t4 ], [t5 + t6 ] 2 2

.

Then I satisfies ( I1a ) − ( I2 ) with ϕ = ψ. The following result is essentially contained in Berinde and Vetro [38]: Theorem 5.2.1. [38]Let ( M, d, ) be a complete ordered metric space and (S, T ) a pair of self-mappings on M such that S( M) ⊆ T ( M) and S is T-increasing. Assume that there exists a function I ∈ IB satisfying (i1a ), such that for all u, v ∈ M with Tu Tv, I (d(Su, Sv), d( Tu, Tv), d( Tu, Su), d( Tv, Sv), d( Tu, Sv), d( Tv, Su)) ≤ 0.

(5.1)

If the following conditions hold: (a1 ) there exists u0 ∈ M such that Tu0 Su0 , (a2 ) for every increasing sequence { Tun } in M converges to Tu, we have Tun Tu,

∀ n ∈ N0 and Tu T ( Tu), then the pair (S, T ) has a coincidence point in M. Moreover, if (a3 ) (S, T ) is weakly compatible pair, (a4 ) I satisfies (i2 ), then the pair (S, T ) has a common fixed point.

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C HAPTER 5: R ESULTS U NDER AN I MPLICIT F UNCTION The authors in [38], also, gave the following sufficient conditions for the

uniqueness of the common fixed point in above theorem: (a5 ) ( M, ) is (S, T )-directed, (a6 ) I satisfies (i1c ). The main results of this chapter are based on the following motivations and observations. (i) To provide an example which shows that Theorem 5.2.1 is not correct in its present form. (ii) To modify Theorem 5.2.1, we employ the completeness of any subspace E (such that S( M ) ⊆ E ⊆ T ( M )) rather than the completeness of whole space M. This point is very vital and also responsible for the failure of Theorem 5.2.1. Moreover, we consider a relatively larger class of implicit functions which also cover some nonlinear contractions besides weakening some earlier metrical notions, such as completeness and continuity. Further, the condition

( a2 ) is replaced by relatively weaker notion so called I-regularity. (iii) To prove a fixed point theorem under a relatively weaker condition of Daneštype (see [61, Definition 1]) which can be viewed as an order analogue version ´ c [55, Theorem 1] for quasi contraction. of a famous theorem due to Ciri´ (iv) To prove a sharper version of Theorem 1 due to Berinde and Vetro [38] in the setting of metric spaces. (v) To provide an application for the newly proved results.

5.3

Common Fixed Point Results

Firstly, we utilize the following example which exhibits that Theorem 5.2.1 is not correct in its present form. Example 5.3.1. Consider M = {u0 , u1 , u2 , ..., un , ...} where u0 = 0, ui = −( 14 )i , i =

{1, 2, 3, ...} with usual metric and usual order. Then ( M, d, ≤) is an ordered metric space. Define two self-mappings S and T on M by: S(ui ) = ui+2 and T (ui ) = ui+1 , for all i.


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Consider the function I ∈ IB defined by [37, Example 3.1]: I (t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − kt2 , where k ∈ [0, 1). With a view to verify assumption (5.1) of Theorem 5.2.1, consider ui , u j in M with i < j so that Su j − Sui ≤ k Tu j − Tui , i.e., 1 i +2 4

−

1 j +2 4

≤k

h 1 i+1 4

−

1 j +1 i 4

,

or 1 i +2 h 4 which means

1 4

1−

1 j −i i 4

≤k

1 i +1 h 4

1−

1 j −i i 4

,

≤ k. Hence, the function I satisfies (i1a ), (i1c ) and (i2 ) for k ∈ [ 41 , 1). Also,

all other assumptions of Theorem 5.2.1 are satisfied. Observe that the pair (S, T ) has no common fixed point. In fact the pair does not admit even a coincidence point. The following notions will be used in our subsequent discussions: Definition 5.3.1. Let ( M, d, ) be an ordered metric space and S, T : M → M such that S( M) ⊆ T ( M ). For every u0 ∈ M, consider the sequence {un } ⊂ M defined by Sun = Tun+1 , for all n ∈ N0 . Then { Tun } is called T-S-sequence with initial point u0 . Notice that the ordered metric space ( M, d, ) is said to be regular w.r.t. T if every increasing sequence { Tun } in M converges to Tu, admits a subsequence

{ Tunk } such that each term of { Tunk } is comparable with Tu. With a view to correct and enrich Theorem 5.2.1, we frame the following: Definition 5.3.2. Let ( M, d, ) be an ordered metric space and T : M → M. We say that

( M, d, ) is (i ) I-regular w.r.t. T if every increasing sequence { Tun } in M converges to Tu, admits a subsequence { Tunk } such that each term of { Tunk } is comparable with Tu and Tu T ( Tu).

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(ii ) D-regular w.r.t. T if every decreasing sequence { Tun } in M converges to Tu, admits a subsequence { Tunk } such that each term of { Tunk } is comparable with Tu and Tu T ( Tu).

(iii ) M-regular w.r.t. T if it is both I-regular and D-regular. Now we are equipped to state and prove our main result as follows: Theorem 5.3.1. Let ( M, d, ) be an ordered metric space and E an O-complete subspace of M. Also, let S, T : M → M be such that S( M) ⊆ E ⊆ T ( M ) and S is T-increasing. Assume that there exists a function I ∈ I satisfying ( I1a ) and such that, for all u, v ∈ M (with Tu Tv), I (d(Su, Sv), d( Tu, Tv), d( Tu, Su), d( Tv, Sv), d( Tu, Sv), d( Tv, Su)) ≤ 0.

(5.2)

If the following conditions hold: (b1 ) there exists u0 ∈ M such that Tu0 Su0 , (b2 ) ( E, d, ) is I-regular w.r.t. T, then the pair (S, T ) has a coincidence point in M. Also, if (b3 ) I satisfies ( I2 ), (b4 ) (S, T ) is weakly compatible pair, then the pair (S, T ) has a common fixed point. Moreover, if (b5 ) C (S, T ) is (S, T )-directed, (b6 ) I satisfies ( I1c ), then the common fixed point is unique. Proof. The proof is divided into three steps as follows: Step 1. For u0 with Tu0 Su0 , we can construct a S-T-sequence {Sun } with initial point u0 satisfying Tu0 Su0 = Tu1 Su1 = Tu2 ... = Tun Sun = Tun+1 Sun+1 ... . Clearly, { Tun }, {Sun } ⊂ S( M) ⊆ E. Moreover, both the sequences are increasing sequences. If Sum = Sum+1 for some m ∈ N, then um+1 is the required coincidence


77

point and we are through. Henceforth, we assume that Sun 6= Sun+1 for all n. As Tun Tun+1 , we can take u = un and v = un+1 in (5.2) so that I (d(Sun , Sun+1 ), d(Sun−1 , Sun ), d(Sun−1 , Sun ), d(Sun , Sun+1 ), d(Sun−1 , Sun+1 ), d(Sun , Sun )) ≤ 0. Since I is decreasing in the fifth variable, on using the triangular inequality, above inequality become I (d(Sun , Sun+1 ), d(Sun−1 , Sun ), d(Sun−1 , Sun ), d(Sun , Sun+1 ), d(Sun , Sun+1 ) + d(Sun−1 , Sun ), 0) ≤ 0. Thus, there exists a Matkowski function ϕ such that d(Sun , Sun+1 ) ≤ ϕ(d(Sun−1 , Sun )). Since ϕ is increasing function, on using induction on n in (5.3), we get d(Sun , Sun+1 ) ≤ ϕn (d(Su0 , Su1 )), for all n ∈ N0 . Let e be fixed. Choose n ∈ N0 so that d(Sun+1 , Sun ) < e − ϕ(ε). Now, d(Sun+2 , Sun ) ≤ d(Sun+2 , Sun+1 ) + d(Sun+1 , Sun )

< ϕ(d(Sun+1 , Sun )) + e − ϕ(ε) ≤ ϕ(e − ϕ(ε)) + e − ϕ(ε) ≤ ϕ(e) + e − ϕ(ε) = ε. Also, d(Sun+3 , Sun ) ≤ d(Sun+3 , Sun+1 ) + d(Sun+1 , Sun )

< ϕ(d(Sun+2 , Sun ) + e − ϕ(ε) ≤ ϕ(e) + e − ϕ(ε) = ε. By induction d(Sun+k , Sun ) < ε, for all k ∈ N

(5.3)

78


so that {Sun } is a Cauchy sequence in the O-complete subspace E. Therefore, there exist w ∈ E and u ∈ M such that w = Tu with Sun ↑ Tu and Tun ↑ Tu.

(5.4)

Step 2. Since ( E, d, ) is I-regular, there exists a subsequence { Tunk } of { Tun } such that Tunk ≺ Tu, ∀ k ∈ N. On putting u = unk , v = u in (5.2), one gets I (d(Sunk , Su), d( Tunk , Tu), d( Tunk , Sunk ), d( Tu, Su), d( Tunk , Su), d( Tu, Sunk )) ≤ 0. As I is a continuous, letting k → ∞ and using (5.4), we get I (d( Tu, Su), 0, 0, d( Tu, Su), d( Tu, Su), 0) ≤ 0, implying thereby d( Tu, Su) ≤ ϕ(0) = 0 so that Tu = Su. Step 3. Since the pair (S, T ) is weakly compatible and Tu = Su(= w for some w ∈ M), we have Tw = T (Su) = S( Tu) = Sw

(5.5)

By assumption (b2 ), Tu TTu = Tw. So, by putting u = u and v = w in (5.2), we get I (d(Su, Sw), d( Tu, Tw), d( Tu, Su), d( Tw, Sw), d( Tu, Sw), d( Tw, Su)) ≤ 0 so that d(Su, Sw) = 0 which along with (5.5) gives rise Tw = Sw = Su = w, i.e., w is a common fixed point of S and T.

Now, we show that the pair (S, T ) has a unique common fixed point in the presence of conditions (b5 ) and (b6 ). Let w and z be two common fixed points of the pair (S, T ). By the (S, T )-directedness of C (S, T ), there exists some t0 ∈ M such that Sz ≺ Tt0 and Sw ≺ Tt0 . Since S( M) ⊆ T ( M ) and S is T-increasing, we can define a sequence {tn } ⊂ M with Ttn+1 = Stn ,


79

and Tz ≺ Ttn , ∀ n ∈ N0 . On setting u = tn , z = z in (5.2), we have 0 ≥ I (d(Stn , Sz), d( Ttn , Tz), d( Ttn , Stn ), d( Tz, Sz), d( Ttn , Sz), d( Tz, Stn ))

= I (d(Stn , Sz), d(Stn−1 , Sz), d(Stn−1 , Stn ), 0, d(Stn−1 , Sz), d(Sz, Stn )) ≥ F d(Stn , Sz), d(Stn−1 , Sz), d(Stn−1 , Sz) + d(Sz, Stn ), 0, d(Stn−1 , Sz), d(Sz, Stn ) so that d(Stn , Sz) ≤ ϕ(d(Stn−1 , Sz)), for the Matkowski function ϕ. On using argument similar to that in Step 1, we can prove that d(Stn , Sz) ≤ ϕn (d(St0 , Sz)), for all n, which on letting n → ∞ on both sides, gives rise d(Stn , Sz) → 0 as n → ∞. Similarly, we can prove that d(Stn , Sw) → 0 as n → ∞. Hence, d(w, z) = d(Sw, Sz) ≤ d(Sw, Stn ) + d(Stn , Sz) → 0 as n → ∞, which amounts to saying that w = z. A comprehension of Theorems 5.3.1 and 5.2.1, we reveals the following facts: • The completeness in Theorem 5.2.1 is merely required on any subspace E rather than the whole space M such that S( M ) ⊆ E ⊆ T ( M). This point is very vital and is also responsible for the failure of the Theorem 5.2.1. • The class of the implicit function utilized in Theorem 5.3.1 is relatively larger than the one utilized in Theorem 5.2.1. • The property embodied in condition ( a2 ) of Theorem 5.2.1 implies the Iregularity (utilized in Theorem 5.3.1). • The notions on ‘continuity and completeness’ employed in Theorem 5.3.1 are relatively weaker than their correspondence notions in Theorem 5.2.1.

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C HAPTER 5: R ESULTS U NDER AN I MPLICIT F UNCTION In fact, we can replace the I-regularity of ( E, d, ) together with condition ( I1c )

from the hypothesis of Theorem 5.3.1 at the cost of the comparability of one of S and T along with a stronger condition on the set C (S, T ), as we shall see in Theorem 5.3.2.

The following three conditions will be utilized in our forthcoming results: (i) S is ( T, O)-continuous. (ii) (S, T ) is an O-compatible pair and both S and T are O-continuous. (iii) S and T are continuous. Theorem 5.3.2. Theorem 5.3.1 remains true if assumptions (b2 ), (b5 ) and (b6 ) are resp. replaced by the following conditions besides retaining the rest of the hypothesis: (c2 ) any one of the assumptions (i), (ii) and (iii) is satisfied. (c5 ) C (S, T ) is totally ordered set. (c6 ) one of S and T is a comparable mapping. Proof. The proof is divided into three steps where step 1 is the same as in the proof of Theorem 5.3.1 and hence omitted. The other two steps are discussed separately as follows: Step 2. Using the conditions embedded in assumption (c2 ), the modified form of Step 2 runs as follows: (i): Since S is ( T, O)-continuous and T (un ) ↑ Tu, we have Sun ↑ Su. Now, owing to the uniqueness of the limit and (5.4), we have Tu = Su. Thus, we are through. (ii): In view of (5.4) and the O-continuity of both S and T, we have lim T (Sun ) = n→∞

Tw, and lim S( Tun ) = Sw. Now, the O-compatibility of the pair (S, T ) gives rise, n→∞

Tw = Sw. (iii): The proof runs on the lines of the proof of Theorem 1 in [15]. We reproduce it here for convenience of the readers. Since S and T are O-continuous, owing to Lemma 1.3.3, there exists a subset A ⊆ M such that T ( A) = T ( M ) and T : A → M is one-one. Without loss of generality, we can choose A such that u ∈ A. Now, define S : T ( A) → T ( M ) by S( Ta) = Sa, ∀ Ta ∈ T ( A), where a ∈ A.

(5.6)


81

As T : A → M is one-one and S( M ) ⊆ T ( M), S is well defined. Since {un } ⊂ M and T ( M) = T ( A), there exists { an } ⊂ A such that Tun = Tan , ∀ n ∈ N0 . By using Lemma 1.3.2, we get Sun = San , ∀ n ∈ N0 . Therefore, owing to (5.4), we have Tan = San ↑ Tu.

(5.7)

On using (5.6), (5.7) and continuity of S, we get Su = S( Tu) = S( lim Tan ) = lim S( Tan ) = lim San = Tu, n→∞

n→∞

n→∞

i.e., Su = Tu. Step 3. As the pair (S, T ) is a weakly compatible pair, we have Tw = T (Su) = S( Tu) = Sw. By the assumption (c5 ), Tu ≺ Tw. Now, on setting u = u and v = w in (5.2), we get 0 ≥ I (d(Su, Sw), d( Tu, Tw), d( Tu, Su), d( Tw, Sw), d( Tu, Sw), d( Tw, Su))

= I (d(Su, Sw), d(Su, Sw), d(Su, Su), d(Sw, Sw), d(Su, Sw), d(Sw, Su)). Now, assumption (I2 ) implies d(Sw, Su) = 0 and hence Tw = Sw = Su = w. Next, we show that the common fixed point w is unique. Let v, w ∈ M be two common fixed points of the pair (S, T ). By repeating earlier arguments, we have Sv = Sw. Thus, the pair (S, T ) has a unique common fixed point. Remark 5.3.1. When the condition (c2 ) is satisfied with (ii ) in Theorem 5.3.2, Remark 1.3.3 implies that (S, T ) is weakly compatible pair. Thus, assumption (b4 ) is not required any more in the hypothesis. The following example exhibits that Theorem 5.3.2 is genuinely different to Theorem 5.3.1. Example 5.3.2. Consider M = (−1, 1] equipped with usual metric. Then ( M, d, ) is an ordered metric space wherein for u, v ∈ M, u v ⇔ (u ≤ v, v 6= 1) or (u = v = 1). Herein ’≤’ stands of the usual order on R.

82


Set T = I M and define S : M → M by Su = u/3, for all u ∈ M. Consider I ∈ I given in Example 5.2.5 so that I satisfies ( I1a ) and ( I2 ) for the Matkowski function ψ(t) = kt, for some k ∈ (0, 1). Thus, (by taking E = M) Theorem 5.3.2 (with assumption (ii)) ensures the existence of a unique common fixed point (namely u = 0). Observe that, Theorem 5.3.1 is not applicable in the context of Example 5.3.2 because the function I does not satisfy condition ( I1c ) as well as ( M, d, ) is not I-regular. It is worth mentioning here that Theorem 5.2.1 is not applicable to present example due to the involvement of relatively weaker completeness notion.

Though, the succeeding two theorems are similar (to Theorems 5.3.1 and 5.3.2), yet there do exist instances wherein the following two theorems are applicable but Theorems 5.3.1 and 5.3.2 are not, which substantiate the utility of such theorems. Theorem 5.3.3. Let M, E, S and T be defined as in Theorems 5.3.1. Assume that there exists a function I ∈ I satisfying conditions ( I1a ) and (5.2). Suppose that the following conditions hold: (d1 ) there exists u0 ∈ M such that Tu0 Su0 , (d2 ) ( E, d, ) is I-regular, then the pair (S, T ) has a coincidence point in M. Also, if (d3 ) (C (S, T ), ) is (S, T )-directed, (d4 ) I satisfies ( I1b ), then the pair (S, T ) has a unique point of coincidence. And if (d5 ) one of S and T is one-one, then the pair (S, T ) has a unique coincidence point. Moreover if (d6 ) (S, T ) is weakly compatible pair, then the pair (S, T ) has a unique common fixed point.


83

Proof. The proof is divided into five steps where Step 1 and Step 2 are the same as in the proof of Theorem 5.3.1 and hence omitted. Other steps run as follows: Step 3. Let u, v, u, v ∈ M be such that Tu = Su = u and Tv = Sv = v.

(5.8)

We assert that u = v. Due to the (S, T )-directedness of M, there exists t0 ∈ M such that Tu ≺ Tt0 and Tv ≺ Tt0 . Now, for Tu ≺ Tt0 , we can define a sequence

{tn } ⊂ M with Ttn+1 = Stn ,

(5.9)

and Tu ≺ Ttn , ∀ n ∈ N0 . We claim that lim d(Su, Stn ) = 0. Two cases arise: n→∞

Firstly, if d(Su, Stm ) = 0 for some m ∈ N0 , then by (5.8) and (5.9), we get d( Tu, Ttm+1 ) = 0. Consequently, by Lemma 1.3.2, we must have, d(Su, Stm+1 ) = 0. By induction on m, we get d(Su, Stn ) = 0, for all n > m. Secondly, suppose that d(Su, Stn ) > 0, for all n ∈ N0 . On putting u = u, v = tn in (5.2) and using assumption (d4 ), we have 0 ≥ I (d(Su, Stn ), d( Tu, Ttn ), d( Tu, Su), d( Ttn , Stn ), d( Tu, Stn ), d( Ttn , Su)

= I (d(Su, Stn ), d(Su, Stn−1 ), 0, d(Stn−1 , Stn ), d(Su, Stn ), d(Stn−1 , Su)) ≥ I (d(Su, Stn ), d(Su, Stn−1 ), 0, d(Su, Stn ) + d(Su, Stn−1 ), d(Su, Stn ), d(Stn−1 , Su)) so that there exists a Matkowski function ϕ with d(Su, Stn ) ≤ ϕ(d(Su, Stn−1 )).

(5.10)

Since ϕ is increasing function, owing to the induction on n [in (5.10)], we have d(Su, Stn ) ≤ ϕn (d(Su, Su0 )), for all n ∈ N0 . Letting n → ∞ on the both sides, we find d(Su, Stn ) → 0. Thus, in all, the claim is established. Similarly, for Tv ≺ Tt0 , one can show that lim d(Sv, Stn ) = 0. n→∞

84

C HAPTER 5: R ESULTS U NDER AN I MPLICIT F UNCTION Now, d(u, v) = d(Su, Sv)

≤ d(Su, Stn ) + d(Stn , Sv) → 0 as n → ∞. Thus, the pair (S, T ) has a unique point of coincidence. Step 4. Let S be one-one. On contrary, assume that there exist two coincidence points u, v ∈ M such that Tu = Su = Tv = Sv. As S is one-one, we have u = v. The similar arguments carries over in case T is one-one. Step 5. Let u, u ∈ M be such that Tu = Su = u. By Lemma 1.2.1, u itself is a coincidence point. In view of step 4, we must have u = u and hence we are through. Theorem 5.3.4. Theorem 5.3.3 remains true if the condition (d2 ) is replaced by any one of the conditions (i), (ii) and (iii) (besides retaining the rest of the hypotheses). Proof. In the proof of the theorem, Steps 1 and 2 are the same as in the proof of Theorem 5.3.2 while Steps 3, 4 and 5 are the same as in the proof of Theorem 5.3.3. Remark 5.3.2. One can obtain dual type results corresponding to all preceding theorems by replacing “O-analogues” with “O-analogues”, the “I-regularity” with “D-regularity” and the condition “Tu0 Su0 ” with “Tu0 Su0 ”. Remark 5.3.3. One can obtain a companied type result corresponding to all preceding theorems by replacing “O-analogues” with “O-analogues”, the “I-regularity” with “Mregularity” and ‘Tu0 Su0 ” with Tu0 ≺ Su0 ”. All proved results of this chapter unify, extend and generalize many relevant common fixed point results from the existing literature which can not be completely ´ c mentioned here. But as an example, we consider a famous theorem due to Ciri´ [55, Theorem 1] and extend the same to a pair of self-mappings satisfying Daneštype contraction in an ordered metric space.


85

Corollary 5.3.1. Let M, E, S and T be defined as in Theorems 5.3.1. Suppose there exists a continuous half-Matkowski function ρ such that, for all Tu Tv, d(Su, Sv) ≤ ρ(max {d( Tu, Tv), d( Tu, Su), d( Tv, Sv), d( Tu, Sv), d( Tv, Su)}). If the following conditions hold:

(e1 ) there exists u0 ∈ M such that Tu0 Su0 , (e2 ) ( E, d, ) is I-regular (resp. any one of the assumptions (i), (ii) and (iii) is satisfied), then the pair (S, T ) has a coincidence point in M. Also, if

(e3 ) (S, T ) is weakly compatible pair, (e4 ) (C (S, T ), ) is (S, T )-directed (resp. C (S, T ) is totally ordered set and one of S and T is comparable), then the pair (S, T ) has a unique common fixed point. Proof. The result is obtained from Theorem 5.3.1 (resp. Theorem 5.3.2) by taking the function I ∈ I defined in Example 5.2.2. Notice that Corollary 5.3.1 can not be derived using Theorem 5.2.1. Remark 5.3.4. Similarly, one can obtain the dual of Corollary 5.3.1 corresponding to Theorems 5.3.3 and 5.3.4. Observe that, setting ρ(t) = kt, k ∈ [0, 12 ) in Corollary 5.3.1, gives rise a linear form of the corollary. Interestingly, we show that this linear form is not valid for k ≥ 12 . Consequently, Corollary 5.3.1 is not valid for a general Matkowski function. Example 5.3.3. Consider M = [0, ∞) with usual metric and usual order. Then ( M, d, ≤) is an ordered metric space. Define S, T : M → M by Su = u2 + 1 and Tu =

2 u, ∀ u ∈ M. 3

By a routine calculation, one can verify that all the conditions of Corollary 5.3.1 are satisfied with k ≥ 12 . Nevertheless, the pair (S, T ) has no coincidence in M. For, if u is a coincidence point, we must have a real root for 3u2 − 2u + 3 = 0 which is not true. In view of Remark 1.3.2, we can deduce the following result:

86


Corollary 5.3.2. Let ( M, d, ) be a complete ordered metric space and S an increasing mapping on M. Suppose there exists a continuous half-Matkowski function ρ such that for all u v, d(Su, Sv) ≤ ρ(d(u, v)). Then S has a fixed point if the following conditions hold: (b1 ) there exists u0 ∈ M such that u0 Su0 , (b2 ) ( M, d, ) is I-regular. Proof. The result is obtained from Theorem 5.3.1 by taking the function I ∈ I defined in Example 5.2.4 and setting T = I M .

5.4

Corresponding Results on Metric Spaces

We can deduce the following sharpened version of Theorem 1 due to Berinde and Vetro [38]. Theorem 5.4.1. Let ( M, d) be a metric space and E a complete subspace of M. Let (S, T ) be a pair of self-mappings on M such that S( M) ⊆ E ⊆ T ( M ). Assume that there exists a function I ∈ I satisfying ( I1a ) such that, for all u, v ∈ M I (d(Su, Sv), d( Tu, Tv), d( Tu, Su), d( Tv, Sv), d( Tu, Sv), d( Tv, Su)) ≤ 0. Then the pair (S, T ) has a coincidence point in M. Moreover, if (S, T ) is weakly compatible pair and I satisfies ( I2 ), then the pair (S, T ) has a unique common fixed point. Proof. The proof is omitted as it is similar to the one given in [38, Theorem 1], except some minor changes corresponding to the new implicit function. Corollary 5.4.1. Let M, E, S and T be defined as in Theorem 5.4.1. Assume that there exists a continuous half-Matkowski function ρ such that, for all u, v ∈ M d(Su, Sv) ≤ ρ(max {d( Tu, Tv), d( Tu, Su), d( Tv, Sv), d( Tu, Sv), d( Tv, Su)}). Then the pair (S, T ) has a unique common fixed point.


87

Remark 5.4.1. Setting ρ(t) = kt (where k ∈ [0, 1/2)]), T = I M and E = S( M) in ´ c Corollary 5.4.1, we reduces it to a partially sharpened version of Theorem 1 due to Ciri´ [55]. Remark 5.4.2. One can drive dual type results corresponding to the results of this section as indicated in Remarks 5.3.2 and 5.3.3.

5.5

An Application

Inspired by [125], we establish the existence of a solution for the following Volterra type integral equation: f (t) =

Z 1 0

K (t, s, f (s))ds + g(t),

t ∈ J = [0, 1],

(5.11)

where K : J × J × R → R and g : J → R are continuous functions. Let C( J ) denotes the space of all continuous functions defined on J. Set d(u, v) = ku − vk∞ = max |u(t) − v(t)|, ∀u, v ∈ C( J ). t∈ J

Then (C( J ), d) is a complete metric space, see [25, 130]. Let (C( J ), d) be equipped with the partial order given by: u, v ∈ C( J ), u v ⇔ u(t) ≤ v(t), for all t ∈ J. Define a function S : C( J ) → C( J ) by Su(t) =

Z 1 0

K (t, s, u(s))ds + g(t), ∀ t ∈ J.

(5.12)

Clearly, if u ∈ C( J ) is a fixed point of S, then u ∈ C( J ) is a solution of (5.11). Consider the following conditions: (a) for all t, s ∈ J and f ∈ C( J ), we have K (t, s, f (t)) ≤ K t, s,

Z 1 0

! K (s, τ, f (τ ))dτ + g(s) ,

(b) there exist a continuous function ω : J 2 → [0, ∞) and a half-Matkowski function ρ such that for all t ∈ J and for all a, b ∈ [0, ∞) with a ≤ b ,

|K (t, s, a) − K (t, s, b)| ≤ ω (t, s)ρ(|b − a|) where ρ is as in Corollary 5.3.2.

88


(c) sup t∈ J

R1 0

ω (t, s)ds ≤ 1.

Now, we prove the following result on the existence of a solution of problem (5.11). Theorem 5.5.1. Problem (5.11) has at least one solution u∗ ∈ C provided that the conditions (a)-(c) are satisfied. Proof. From (a), for all t ∈ J, we have Su(t) =

≤

Z 1 0

Z 1 0

=

Z 1 0

K (t, s, u(s)ds + g(t) K t, s,

Z 1 0

!

)K (s, τ, u(τ ))dτ + g(s) ds + g(t)

K (t, s, Su(s)ds + g(t)

= S(Su)(t).

(5.13)

Therefore, Su ≤ S(Su) for all u ∈ C( J ) so that condition (b1 ) of Corollary 5.3.2 is satisfied. Now, for all u, v ∈ C( J ) with v u, by (b) and (c), we have Z 1 Z 1 |Su(t) − Sv(t)| = K (t, s, u(s))ds − K (t, s, v(s))ds 0 0 Z 1 ≤ K (t, s, u(s) − K (t, s, v(s) ds 0

≤

Z 1 0

≤

Z 1 0

ω (t, s)ρ(|u(s) − v(s)|)ds ω (t, s)ρ(d(u, v))ds

≤ ρ(d(u, v)). Thus, d(Su, Sv) ρ(d(u, v)) for all u, v ∈ C( J ) with v u. Also, it is proved in [128] that for every increasing sequence {Sun } in C( J ) converges to Su, there exists a subsequence {Sunk } of {Sun }} such that each term of {Sunk } is comparable with Su. Further, by (5.13) we have Su S(Su) so that (C( J ), d, ) is I-regular. Finally, Corollary 5.3.2 ensures the existence of some u∗ ∈ C such that u∗ = Su∗ i.e, u∗ is a solution of Volterra type integral equation given in (5.11).

CHAPTER

Results Under F Function

6

We observe that some results involving α-type F- contractions of the existing literature are not correct in their present forms. Starting from this point, in this chapter, we prove some order-theoretic fixed point results for extended F-weak contraction. Our observations and the usability of our results are substantiated using suitable examples. As an application, we prove an existence and uniqueness result for the solution of a first-order ordinary differential equation satisfying periodic boundary conditions in the presence of either its lower or upper solution.

6.1


Recall that the auxiliary function F is a mapping F : (0, ∞) → R satisfying the following three conditions: F1: F is strictly increasing, F2: for every sequence {sn } of positive real numbers, lim sn = 0 ⇔ lim F (sn ) = −∞,

n→∞

n→∞

F3: there exists k ∈ (0, 1) such that lim sk F (s) = 0. s →0+

Let us denote by F the family of all functions F satisfying conditions (F1)-(F3). The content of this chapter is based on the following article: M. Imdad, R. Gubran, M. Arif and D. Gopal An observation on α-type F-contractions and some ordered-theoretic fixed point results Math. Sci., 11 (3) 2017, 247–255.

89

90

C HAPTER 6: R ESULTS U NDER F F UNCTION Utilizing bove type of auxiliary functions, Wardowski [187] generalized Banach

contraction principle in a novel way by introducing a new type of contractions called F-contractions. Definition 6.1.1. [187] Let ( M, d) be a metric space. A mapping S : M → M is said to be an F-contraction if there exists τ > 0 and F ∈ F such that d(Su, Sv) > 0 ⇒ τ + F (d(Su, Sv)) ≤ F (d(u, v)),

(6.1)

for all u, v ∈ M. Theorem 6.1.1. [187] Let ( M, d) be a complete metric space and S : M → M an Fcontraction mapping. Then S possesses a unique fixed point. Interestingly, on varying the elements of F suitably, a variety of known contractions in the literature can be deduced. Here, We mentioned one example that deduce Banach contraction principle. Some other examples can be found in [187] and subsequent articles in the existing literature. Example 6.1.1. [187] Consider F ∈ F given by F (s) = ln s. Then each self-mapping S on a metric space ( M, d) satisfying inequality (6.1) is an F-contraction such that d(Su, Sv) ≤ e−τ d(u, v), where u, v ∈ M and u 6= v. Observe that, this inequality holds trivially if u = v. Secelean [162] gave an equivalent condition to condition (F2). To prove the same, Piri and Kumam [139] replaced the condition (F3) by assuming F to be continuous on (0, ∞). Shukla and Radenović [170] proved some common fixed point theorems ´ c-type) in 0-complete partial metric spaces while Shukla et for F-contractions (of Ciri´ al. [171] proved some fixed point theorems for ordered F-generalized contractions in ordered 0-S-orbitally complete partial metric spaces. Batra et al. [32] extended the concept of F-contraction on a metric space endowed with a graph and generalize some results related with G-contraction for a directed graph G. On employing C’irić-type contractions in Definition 1.2.8 , Wardowski and Van Dung [188] (also independently Minak et al. [119]) introduced the notion of F-weak

C HAPTER 6: R ESULTS U NDER F F UNCTION

91

contraction and utilized the same to generalize Theorem 6.1.1 as well as several other theorems of the existence literature. Definition 6.1.2. [119, 188] Let ( M, d) be a complete metric space. A mapping S : M → M is said to be an F-weak contraction if there exists τ > 0 and F ∈ F such that d(Su, Sv) > 0 ⇒ τ + F (d(Su, Sv)) ≤ F ( MS (u, v)), ∀u, v ∈ M.

(6.2)

Theorem 6.1.2. [119, 188] Let ( M, d) be a complete metric space and S : M → M be an F-weak contraction for some F ∈ F . Then S has a unique fixed point provided (i) F is continuous or (ii) S is continuous. Employing the notion of α-admissible mapping (see Definition 1.2.2), Gopal et al. [72] introduced the concept of α-type F-contraction (in short αF-contraction) as follows: Definition 6.1.3. [72] Let ( M, d) be a complete metric space. A mapping S : M → M is said to be an αF-weak contraction if there exists τ > 0, F ∈ F and α : M × M →

{−∞} ∪ (0, +∞) such that d(Su, Sv) > 0 ⇒ τ + α(u, v) F (d(Su, Sv)) ≤ F ( MS (u, v)), ∀u, v ∈ M.

(6.3)

Employing Definition 6.1.3, Gopal et al. [72] proved the following result. Theorem 6.1.3. [72] Let ( M, d) be a complete metric space and S : M → M an αF-weak contraction. Suppose that the following conditions hold: (i) there exists u0 ∈ M such that α(u0 , Su0 ) ≥ 1, (ii) S is α-admissible, (iii) either S is continuous or (F is continuous and if a sequence {un } ∈ M such that α(un , un+1 ) ≥ 1, ∀n ∈ N and un → u as n → ∞, then α(un , u) ≥ 1). Then S has a fixed point. Definition 6.1.4. [53] Let ( M, d) be a metric space and S : M → M. Then M is said to be S-orbitally complete if every Cauchy sequence {Sn u} ∈ M converges in M.

92

C HAPTER 6: R ESULTS U NDER F F UNCTION One of the earliest order-theoretic fixed point results concerning F-contraction

was due to Sgroi and Vetro [167] wherein the authors proved some fixed point results for closed multi-valued F-contractions of Hardy-Rogers-type. Following this line of research, Cosentino and Vetro [59] obtained some fixed point theorems of Hardy-Rogers-type. Later on, Abbas et al. [1] introduced the idea of F-contraction with respect to a self-mapping to obtain common fixed point results. Very Recently, perhaps knowledgeably, Durmaz et al. [65] proved the following result which can be obtained by setting g = I M in Theorem 2 of [1]: Theorem 6.1.4. [65] Let ( M, , d) be a complete ordered metric space and S : M → M an F-contraction for some F ∈ F . Suppose the following conditions hold: (i) there exists u0 ∈ M such that u0 Su0 , (ii) S is increasing, (iii) either S is continuous (or F is continuous and ( M, d, ) enjoys ICU-property). Then S has a fixed point. Further, the authors in [65] adopted the condition (1.10) to ensure the uniqueness of the fixed point in Theorem 6.1.4: Remark 6.1.1. Very recently, Vetro [185] enlarged the class F (and denote the same F) by withdrawing the condition (F3) and replacing the constant τ by a function σ : (0, ∞) →

(0, ∞) with lim inf σ(t) > 0, ∀s ≥ 0. t→s+

Obviously, F ⊆ F and F (s) = −1/s is a member of F which is not in F . We denote with S the family of all functions σ. Definition 6.1.5. [55] Let ( M, d) be a metric space and S : M → M. Then for arbitrary u ∈ M, O(u) = {Sn u : n = 0, 1, 2, ....} is called the orbit of u. Further, M is said to be S-orbitally complete if and only if every Cauchy sequence which is contained in O(u) converges in M.


6.2

93

An Observation on α-Type F-Contractions

We begin our observation with [72] wherein authors enlarged the co-domain of α to include −∞ and at the same time assumed that the expression −∞.0 has the value

−∞ which is quite unnatural (see Definition 1.2.2). Inspired by this substitution, we were able to furnish the following counterexamples. Example 6.2.1. Let M = {0, 14 , 12 , 1} equipped with usual metric d. Then ( M, d) is a complete metric space. Define α : M × M → {−∞} ∪ (0, ∞) by:     −∞, for u, v ∈ {0, 1}, u 6= v,    α(u, v) = 2 − ln 3 , for u, v ∈ { 1 , 1 }, u 6= v, 4 2 ln 4      1, otherwise. Let S be a self-mapping on M defined as: S0 = 1, S 41 = 12 , S 12 =

1 4

and S1 = 0. Then S is

continuous as well as α-admissible. By a routine calculation, one can verify that S satisfies the contraction condition (6.3) for F (s) = ln s and τ = ln 43 . Especially, for u = 0 and v = 1, we have −∞ = ln 4/3 + (−∞) ln(1) ≤ ln max 1, 1, 1, 0 = 0. Observe that, S is fixed point free which disproves Theorem 6.1.3. Even if we restrict the co-domain of α to (0, ∞) in Definition 6.1.3 with a view to recover Theorem 6.1.3, still the theorem continues to be erroneous. The following example exhibits this fact: Example 6.2.2. Consider M = [1, ∞) equipped with the discrete metric D, that is,   0, for u = v, D (u, v) =  1, otherwise. Take Su = au, ∀u ∈ M where a ∈ (1, ∞). Then with α(u, v) = 2, ∀u, v ∈ M and F (s) = − √1s , S satisfies all the requirements of Theorem 6.1.3 (for τ < 1) but S is a fixed point free.

94

C HAPTER 6: R ESULTS U NDER F F UNCTION Indeed, in all the proofs of the results on αF-contractions, e.g., in [72, line 4, page

962] and also in [132, equation (2.4)], the authors assumed that F ( x ) ≤ α(u, v) F ( x ), for α(u, v) ≥ 1 which is not true in general (as F may have negative value).

6.3

Fixed Point Results

In order to generalize Theorem 6.1.4, the following definition is required: Definition 6.3.1. Let ( M, d) be a metric space. A mapping S : M → M is said to be an extended F-weak contraction if there exist σ ∈ S and F ∈ F such that d(Su, Sv) > 0 ⇒ σ(d(u, v)) + F (d(Su, Sv)) ≤ F ( MS (u, v)),

(6.4)

for all u, v ∈ M with u v. Theorem 6.3.1. Let ( M, , d) be an ordered metric space and S : M → M an extended F-weak contraction and ( M, d) is S-orbitally complete metric space. Suppose that the following conditions hold: (i) there exists u0 ∈ M such that u0 Su0 , (ii) S is increasing, (iii) F is continuous and ( M, d, ) enjoys ICC property. Then S has a fixed point u ∈ M. Moreover, for every u0 ∈ M satisfying (i), the sequence

{Sn u0 } converges to u. Proof. Let u0 ∈ M be such that u0 Su0 . Define a sequence {un } in M by un+1 := Sun , ∀n ∈ N0 := N ∪ {0}. If un = un+1 for some n ∈ N0 , then we are done. Otherwise, we assume d(un , un+1 ) > 0, ∀n ∈ N . As u0 Su0 and S is increasing, we have u0 u1 u2 ... un un+1 ... . Now, on setting u = un−1 and v = un in (6.4), we have σ(d(un−1 , un )) + F (d(un , un+1 )) ≤ F ( MS (un−1 , un )) = F max {d(un−1 , un ), d(un , un+1 )} .


95

If d(un−1 , un ) ≤ d(un , un+1 ), for some n ∈ N, then F (d(un , un+1 )) ≤ F (d(un , un+1 )) − σ (d(un−1 , un )), a contradiction as σ (d(un−1 , un )) > 0. Therefore, F (d(un , un+1 )) ≤ F (d(un−1 , un ) − σ (d(un−1 , un )) which, in turn, yields F (d(un , un+1 )) ≤ F (d(u0 , u1 )) − n inf{σ (d(un−1 , un ))}, n

(6.5)

for all n. On letting n → ∞ in (6.5), we get lim F (d(un , un+1 )) = −∞. Therefore, due to n→∞

(F2), lim d(un , un+1 ) = 0.

n→∞

(6.6)

We assert that {un } is a Cauchy sequence . Let us assume that {un } is not so. Then there exist e > 0 and two subsequences {unk } and {umk } of {un } such that nk > mk ≥ k, d(unk , umk ) ≥ e and d(unk −1 , umk ) < e, ∀k ∈ N. Now, we have e ≤ d ( u n k , u m k ) ≤ d ( u n k , u n k −1 ) + d ( u n k −1 , u m k ) ≤ d ( u n k , u n k −1 ) + e so that lim d(unk , umk ) = e.

k→∞

Again, we have e ≤ d ( u n k , u n k +1 ) + d ( u n k +1 , u m k +1 ) + d ( u m k +1 , u m k ) so that (on letting k → ∞) e ≤ lim inf d(unk +1 , umk +1 ). k→∞

Similarly, we can deduce that e ≤ lim inf d(unk +1 , umk ) and e ≤ lim inf d(umk +1 , unk ). k→∞

k→∞

It follows that there exists l ∈ N with d(unk +1 , umk +1 ) > 0, d(unk +1 , umk ) > 0 and d(umk +1 , unk ) > 0, ∀k ≥ l. Then, for all k ≥ l, we have (on setting u = unk and

96


v = umk in (6.4)) σ(d(unk , umk )) + F (d(unk +1 , umk +1 )) ≤ F ( MS (unk , unk )), where MS (unk , umk ) = max d(unk , umk ), d(unk , unk +1 ), d(umk , umk +1 ), d ( u n k , u m k +1 ) + d ( u m k , u n k +1 ) 2 ≤ max d(unk , umk ), d(unk , unk +1 ), d(umk , umk +1 ), d ( u n k , u m k ) + d ( u m k , u m k +1 ) + d ( u m k , u n k ) + d ( u n k , u n k +1 ) . 2 Letting k → ∞ in presiding inequality and in view of the definition of σ and the continuity of F, we get F (e) < lim inf σ (d(umk , unk )) + F (e) ≤ F (e), k→∞

a contradiction so that {un } is a Cauchy sequence and having a limit u ∈ M (due to the completeness assumption). Next, we show that u is a fixed point. Suppose that un = Su for infinitely many n ∈ N, then there exists a subsequence of {un } which converges to Su and the uniqueness of the limit finish the proof. Henceforth, we may assume that Sun 6= Su,

∀n ∈ N0 . On using the ICC property of ( M, d, ), there exists a subsequence {unk } of {un } and a positive integer k0 such that unk u, ∀nk ≥ k0 . Now, for nk ≥ k0 , we can set u = unk and v = u in (6.4) so that σ(d(unk , u)) + F (d(unk +1 , Su)) ≤ F ( M(un , u)) n ≤ F max d(unk , u), d(unk , unk +1 ), d(u, Su), o 1 [d(unk , u) + d(u, Su) + d(u, unk +1 )] (. 6.7) 2 Let, on the contrary, d(u, Su) > 0. Making n → ∞ in (6.7), one gets γ + F (d(u, Su)) ≤ F (d(u, Su)), where 0 < γ =

lim inf σ(d(un , u)), a contradiction so that d(u, Su) = 0 which

d(un ,u)→0+

concludes the proof.


97

The following result is yet another version of Theorem 6.3.1 . Theorem 6.3.2. Theorem 6.3.1 remains true if the condition (iii) is replaced by the continuity of S whenever F ∈ F . Proof. The proof is identical to the proof of Theorem 6.3.1 up to (6.6) i.e., lim d(un , un+1 ) = 0.

n→∞

Due to (F3), there exists k ∈ (0, 1) such that lim (d(un , un+1 ))k F (d(un , un+1 )) = 0.

n→∞

(6.8)

Now, from (6.5), we have d(un , un+1 )k [ F (d(un , un+1 )) − F (d(u0 , u1 ))] ≤ −nσ(d(u0 , u1 ))d(un , un+1 )k ≤ 0. (6.9) On using (6.6), (6.8) and letting n → ∞ in (6.9), we get lim nσ (d(u0 , u1 ))d(un , un+1 )k = 0.

n→∞

Hence, there exists m ∈ N0 such that nd(un , un+1 )k ≤ 1, ∀n ≥ m so that d ( u n , u n +1 ) ≤

1

, ∀n ≥ m. (6.10) 1 nk We assert that {un } is a Cauchy sequence. Consider s, t ∈ N0 with s > t ≥ m. Using the triangle inequality and (6.10), we have s −1

d(ut , u x ) ≤

∑ d ( u i , u i +1 )

i =t ∞

≤

∑ d ( u i , u i +1 )

i =t ∞

≤

∑

i =t

As ∑i∞=1

1

1

ik

1 1

ik

.

is convergent, letting s, t → ∞ gives rise lim d(us , ut ) = 0 so that the s,t→∞

assertion is established. Since M is S-orbitally complete, there exists u ∈ M such that lim un = x. The continuity of S implies n→∞

u = lim un+1 = S( lim un ) = Su. n→∞

n→∞

This concludes the proof. Observe that Theorem 6.3.5 carries some advantage over Theorem 6.3.6 as F

98


remains a relatively larger class as compared to F and at the same time most of the utilized functions in F are already continuous. Corollary 6.3.1. Theorem 6.1.4 follows from Theorems 6.3.1 and 6.3.2. The following example exhibits that Theorem 6.3.1 is a proper generalization of Theorem 6.1.4. Example 6.3.1. Let M = A ∪ B ∪ C where A = [0, 1], B = (1, 32 ] and C = ( 23 , 2]. Then

( M, d, ) is an ordered metric space wherein the partial order ’’ on M is defined by: u v ⇔ either u = v or u ≤ y : (u ∈ A and v ∈ B) or (u ∈ B and v ∈ C )}. Consider F ∈ F given by F (s) =

− √1 , s

for s > 0 and σ(t) = 21 , ∀t ∈ (0, ∞). Define a

self-mapping S on M by:    1,    Su = 3 , 2      2,

for u ∈ A, for u ∈ B, for u ∈ C.

Now, in order to verify inequality (6.4), we distinguish the following two cases: Case 1: u ∈ A and v ∈ B. Here, we have n n 3 v − u 1 oo F inf MS (u, v) = F inf max v − u, 1 − u, − v, + 2 2 4 u∈ A,v∈ B u∈ A,v∈ B n n 1 oo 3 = F inf max v − 1, − v, v − 2 4 v∈ B 3 = F 4 2 = −√ . 3 √ Since σ(d(u, v)) + F d(Su, Sv) = 12 + F 12 = 21 − 2, S verifies (6.4). Case 2: u ∈ B and v ∈ C. Here, we have n n 3 v − u 1 oo F inf MS (u, v) = F inf max v − u, u − , 2 − v, + 2 2 4 u∈ B,v∈C u∈ B,v∈C n n oo 3 1 = F inf max v − , 2 − v, v − 2 2 v∈C = F (1)

= −1.

C HAPTER 6: R ESULTS U NDER F F UNCTION Since σ(d(u, v)) + F d(Su, Sv) =

1 2

+F

1 2

99

=

1 2

−

√

2, S verifies (6.4) in this case too.

Therefore, in all, S is an F-contraction ensuring the existence of some fixed point of S. Observe that for u =

−2. As

1 2

3 2

and v = 2, the right hand side of (6.1) gets us F (1/2) =

+ F (d(S(3/2), S2)) =

1 2

− 2, the inequality (1.2.8) does not hold so that

Theorem 6.1.4 is not applicable in the context of present example. Now, we prove the following uniqueness result corresponding to Theorems 6.3.1 and 6.3.2. Theorem 6.3.3. If in addition to the hypotheses of Theorem 6.3.1 (or Theorem 6.3.2), the following condition is satisfied, then S has a unique fixed point. (U) Fix (S) is a totally ordered set. Proof. We prove the conclusion for Theorem 6.3.1 (for Theorem 6.3.2, the proof is similar). If F ∈ F the proof is similar with σ (d(u, v)) ≡ τ. Let u, v be two elements of Fix (S) such that d(u, v) > 0. Then σ (d(u, v)) + F (d(u, v)) ≤ σ(d(u, v)) + F (d(Su, Sv)) d(u, Sv) + d(v, Su) ≤ F max d(u, v), d(u, Su), d(v, Sv), 2 = F (d(u, v)), a contradiction so that d(u, v) = 0. In the following uniqueness result, we weaken the condition (U) at the cost of a relatively more stronger contraction condition. Theorem 6.3.4. If in addition to the hypotheses of Theorem 6.3.1, the directed condition (1.10) is satisfied, then S has a unique fixed point provided MS (u, v) in the contraction condition (6.4) is replaced by mS (u, v). Proof. Let u, v be two elements of Fix (S). There exists w ∈ M such that w is comparable to both u and v. for u ≺ w, we may assume that w u (similar arguments for v ≺ w). Since S is increasing, we deduce that Sn w u, Sn w v.

100


Let tn := d(u, Sn w). We assert that lim tn = 0. For substitution u = u, v = Sn w in n→∞

the contraction condition, we have F ( t n +1 ) < σ ( t n ) + F ( t n +1 )

(6.11)

≤ F (mS (u, Sn w)) 0 + d(Sn w, Sn+1 w) tn+1 + tn = F max tn , , 2 2 t + tn = F max tn , n+1 . 2

(6.12)

Now, if tn < tn+1 , then (6.12) becomes

+ tn F ( t n +1 ) < F , 2 and since F is strictly increasing, we have tn+1 ≤ tn which is a contradiction. t

n +1

Therefore, tn ≥ tn+1 so that tn is a decreasing sequence of nonnegative reals such that lim tn = r ≥ 0. If r > 0, then on letting n tends to infinity in (6.11), we n→∞

get F (r ) < F (r ) which is not possible. Thus, in all situations, lim d(u, Sn w) = n→∞

0. Similarly, we can prove that lim

n→∞

d(v, Sn w)

= 0. Since d(u, v) ≤ d(u, Sn w) +

d(Sn w, v) → 0 as n → ∞, the uniqueness of the fixed point is established. This concludes the proof. Remark 6.3.1. As 1 and 2 are not comparable elements, in the context of Example 6.3.1, the fixed point is not unique supporting our uniqueness results. Observe that by widening the class of functions F in Definition 6.1.2, one can derive following result which remains a metric-version of Theorem 6.3.1. Theorem 6.3.5. Let ( M, d) be a metric space and S : M → M an extended F-weak contraction for some function F ∈ F. If ( M, d) is S-orbitally complete and the following condition holds: (i) F is continuous. Then S has a unique fixed point u ∈ M. Moreover, for every u ∈ M, the Picard sequence

{Sn u} converges to u.


101

Proof. The proof of existence part is very similar to that one of Theorem 6.3.1 and the uniqueness follows from Theorem 6.3.3. Only we mention here that the extra conditions therein ensure the comparability between the element, in which we apply to inequality (6.4). Remark 6.3.2. With a view to check the validity of Theorem 6.3.5 in the context of Example 6.3.1 (without any partial order on M), observe that for u = 1 and v = 2, (6.4) gives rise 1 1 − = + F (d(S1, S2)) ≥ F (1) = −1 2 2 so that the inequality (6.4) is not satisfied. This demonstrates the utility of proving an ordered version of Theorem 6.3.5. The following is yet another version of Theorem 6.3.5 which remains a slightly sharpened form of Theorem 6.1.2 (proved for continuous mapping S). Theorem 6.3.6. Let ( M, d) be a metric space and S : M → M an extended F-weak contraction for some function F ∈ F . If ( M, d) is S-orbitally complete and (i) S is continuous. Then S has a unique fixed point u ∈ M. Moreover, for every u ∈ M, the Picard sequence

{Sn u} converges to u. The proof is omitted as it is very similar to that of [179, Theorem 2.4] and [119, Theorem 2.2] where the completeness of the whole space is utilized rather than the completeness of the orbit of S. Corollary 6.3.2. Theorem 6.1.2 follows from Theorems 6.3.5 and 6.3.6. Corollary 6.3.3. Let ( M, d) a complete metric space and S : M → M. Assume that there exist F ∈ F and σ ∈ S such that S is an F-contraction of Hardy-Rogers, i.e., σ(d(u, v)) + F (d(Su, Sv)) ≤ F a1 d(u, v) + a2 d(u, Su) + a3 d(v, Sv) + a4 d(u, Sv) + a5 d(v, Su) , ∀u, v ∈ M, whenever d(Su, Sv) > 0, where ai ∈ [0, ∞) ∀i, a1 + a2 + a3 + 2a4 = 1, a3 6= 1 and a1 + a3 + a5 ≤ 1. Then S has a unique fixed point u ∈ M.

102


Proof. For all u, v ∈ M, we have a1 d(u, v) + a2 d(u, Su) + a3 d(v, Sv) + a4 d(u, Sv) + a5 d(v, Su) d(u, Sv) + d(v, Su) ≤ ( a1 + a2 + a3 + 2a4 ) max d(u, v), d(u, Su), d(v, Sv), 2 d(u, Sv) + d(v, Su) = max d(u, v), d(u, Su), d(v, Sv), 2 Now, the proof is immediate from Theorems 6.3.1 and 6.3.2

6.4

An Application

Inspired by [127], we establish the existence and uniqueness solution for the following first order periodic boundary value problem with respect to its lower or upper solution:   u0 (s) = g(s, u(s)),

s ∈ J = [0, 1]

(6.13)

  u (0) = u (1), where g : J × R → R is a continuous function. Let C( J ) denotes the space of all continuous functions defined on J. We recall the following two definitions: Definition 6.4.1. [127] A function u ∈ C 1 ( J ) is said to be a lower solution of (6.13), if   u0 (s) ≤ g(s, u(s)), s ∈ J   u (0) ≤ u (1). Definition 6.4.2. [127] A function u ∈ C 1 ( J ) is said to be an upper solution of (6.13), if   u0 (s) ≥ g(s, u(s)), s ∈ J   u (0) ≥ u (1). Now, we prove the following result on the existence and uniqueness of solution of the problem described by (6.13) in the presence of a lower solution (or an upper solution).


103

Theorem 6.4.1. Jn respect of the problem (6.13), suppose that the following conditions hold: (i) there exists τ > 0 such that for all u, v ∈ R with u ≤ v 0 ≤ g(s, v) + e−τ v − [ g(s, u) + e−τ u] ≤ e−2τ (v − u).

(6.14)

(ii) there exists a function ω : R2 → R such that, ∀s ∈ J and ∀ a, b ∈ R with ω ( a, b) ≥ 0, Z 1 ω G (s, t)[ g(t, u(t)) + e−τ u(t)]dt, u(s) ≥ 0 0

where u ∈

C 1( J)

is a lower solution of (6.13).

(iii) for all s ∈ J and all u, v ∈ C 1 ( J ), ω (u(s), v(s)) ≥ 0 implies Z 1 Z 1 ω G (s, t)[ g(t, u(t)) + e−τ u(t)]dt, G (s, t)[ g(t, v(t)) + e−τ v(t)]dt ≥ 0, 0

0

(iv) if un → u ∈

C 1( J)

and ω (un+1 , un ) ≥ 0, then ω (un , u) ≥ 0, ∀n ∈ N.

Then the existence of a lower solution of problem (6.13) ensures the existence and uniqueness of a solution of problem (6.13). Proof. The problem described by (6.13) can be rewritten as:   u0 (s) + e−τ u(s) = g(s, u(s)) + e−τ u(s), ∀ s ∈ J   u (0) = u (1). which is equivalent to the integral equation u(s) =

Z 1 0

G (s, t)[ g(t, u(t)) + e−τ u(t)]dt

where Green function G (s, t) is given by  −τ   ee −(τ1+t−s) , e e −1 G (s, t) = − τ   ee −τ(t−s) , e e

−1

0 ≤ t < s ≤ 1,

(6.15)

.

0 ≤ s < t ≤ 1.

Define a function S : C( J ) → C( J ) by Su(s) =

Z 1 0

G (s, t)[ g(t, u(t)) + e−τ u(t)]dt, ∀ s ∈ J.

(6.16)

Clearly, if u ∈ C( J ) is a fixed point of S, then u ∈ C 1 ( J ) is a solution of (6.15) and hence of (6.13). Now, define a metric d on C( J ) by: d(u, v) = sup |u(s) − v(s)|, s∈ J

∀ u, v ∈ C( J ).

(6.17)

104

C HAPTER 6: R ESULTS U NDER F F UNCTION On C( J ), define a partial order by: u, v ∈ C( J ); u v ⇐⇒ u(s) ≤ v(s), ∀ s ∈ J.

(6.18)

Clearly, (C( J ), d, ) is a complete ordered metric space. We check all other conditions of Theorem 6.3.4: Firstly, let u ∈ C 1 ( J ) be a lower solution of (6.13), we have u0 (s) + e−τ u(s) ≤ g(s, u(s)) + e−τ u(s), ∀ s ∈ J. Multiplying both the sides by ee

(u(s)ee

−τ s

−τ s

, we get

)0 ≤ [ g(s, u(s)) + e−τ u(s)]ee

−τ s

, ∀ s ∈ J,

which implies that u(s)e

e−τ s

≤ u (0) +

Z s 0

[ g(t, u(t)) + e−τ u(t)]ee

−τ t

dt, ∀ s ∈ J.

(6.19)

As u(0) ≤ u(1), we have u (0) e

e−τ

≤ u (1) e

e−τ

≤ u (0) +

so that u (0) ≤

ee

Z 1 0

ee

−τ t

−τ

−1

Z 1 0

[ g(t, u(t)) + e−τ u(t)]ee

−τ t

[ g(t, u(t)) + e−τ u(t)]dt.

dt

(6.20)

On using (6.19) and (6.20), we obtain u(s)e

e−τ s

≤

Z 1 0

≤

Z s 0

ee

−τ t

Z s

−τ t

[ g(t, u(t)) + e−τ u(t)]dt e −1 0 −τ − τ (1+ t ) Z 1 e ee t e −τ [ g(t, u(t)) + e u(t)]dt + [ g(t, u(t)) + e−τ u(t)]dt −τ −τ e e e −1 −1 s e e−τ

[ g(t, u(t)) + e

−τ

u(t)]dt +

ee

so that u(s) ≤

=

Z s e − τ (1+ t − s ) e 0

Z 1 0

−τ ee

−1

[ g(t, u(t)) + e

−τ

u(t)]dt +

Z 1 e−τ (t −s ) e s

−τ ee

−1

[ g(t, u(t)) + e−τ u(t)]dt

G (s, t)[ g(t, u(t)) + e−τ u(t)]dt

= Su(s), for all s ∈ J, which implies that u S(u). Secondly, take u, v ∈ C( J ) such that u v, then by (6.14), we have g(s, u(s)) + e−τ u(s) ≤ g(s, v(s)) + e−τ v(s), ∀ s ∈ J.

(6.21)


105

On using (6.16), (6.21) and the fact that G (s, t) > 0 for (s, t) ∈ J × J, we get Su(s) =

≤

Z 1

G (s, t)[ g(t, u(t)) + e−τ u(t)]dt

0

Z 1

G (s, t)[ g(t, v(t)) + e−τ v(t)]dt

0

= (Sv)(s) ∀ s ∈ J, which owing to (6.18) implies that S(u) S(v) so that S is increasing. Finally, take an increasing sequence {un } ⊂ C( J ) such that un → u ∈ C( J ), then for each s ∈ J, {un (s)} is a sequence in R converging to u(s). Hence, for all n and for all s ∈ J, we have un (s) ≤ u(s), ∀n ∈ N0 so that C( J ) enjoys ICC property. Now we show that S is F-contraction for some F ∈ F. Take u, v ∈ C( J ) such that u v, using (6.14), (6.16) and (6.17), we have d(Su, Sv) = sup |(Su)(s) − (Sv)(s)| = sup (Sv)(s) − (Su)(s) s∈ J

≤ sup s∈ J

≤ sup s∈ J

s∈ J

Z 1 0

Z 1 0

G (s, t)[ g(t, v(t)) + e−τ v(t) − g(t, u(t)) − e−τ u(t)]dt G (s, t)e−2τ (v(t) − u(t))dt

= e−τ d(u, v) sup

Z 1 0

s∈ J

= e−2τ d(u, v) sup s∈ J

= e−τ d(u, v)

G (s, t)dt

1 −τ is 1 1 e −τ ( t − s ) i1 e (1+ t − s ) e + e −τ e−τ 0 s ee − 1 e−τ

−τ 1 ( e e − 1) − 1)

( e e−τ

= e−τ d(u, v) d(u, Su) + d(v, Sv) d(u, Sv) + d(v, Su) −τ ≤ e max d(u, v), , , 2 2 for all u, v ∈ 1 with u v. Hence, S is F-weak contraction for τ chosen as in (i ) and F (s) = ln s. Thus, all the conditions of Theorem 6.3.1 are satisfied ensuring the existence of some fixed point of S. Observe that, for arbitrary u, v ∈ C( J ), w := max{u, v} ∈ C( J ) is comparable to both u and v. Therefore, by Theorem, 6.3.4, S has a unique fixed point which means that problem (6.13) has a unique solution.

106


Theorem 6.4.2. Theorem 6.4.1 remains true if we replace the existence of the lower solution of (6.13) by the existence of an upper solution

CHAPTER

Results Under Simulation Function

7

In 2015, Khojasteh et al. [105] introduced the class of simulation functions and utilized the same to unify several fixed point results of the existing literature. Later on, Karapınar [100] enlarged this class to cover α-admissible contractions. Motivated by aforementioned articles, we establish common fixed point results for α-admissible mappings satisfying a nonlinear contraction condition under simulation function. As an application, we establish the existence of solution for certain two-point boundary value problem of second order differential equation.

Before we go further in this chapter, it is worth mentioning here that Samet et al. [159] succeeded to deduce order-theoretic result using the idea of α-admissible mappings.

7.1

Introduction

It is well known that Banach contraction principle is a fundamental result in fixed point theory and nonlinear analysis. This celebrated principle is very applicable especially in existence and uniqueness problems in mathematics, sciences and some The content of this chapter is based on the following article: R. Gubran, W. M. Alfaqih and M. Imdad Common fixed point results for α-admissible mappings via simulation function, J. Anal., 25 (2) 2017, 281–290.

107

108

C HAPTER 7: R ESULTS U NDER S IMULATION F UNCTION

other domains. Thus far, a multitude of fixed point results enriching this principle has been introduced and utilized. In recent past, several attempts have been made to unify a multitude of existing results, that is, by introducing auxiliary functions in order to prove several theorems in a unified way. Thus, it is possible to treat several fixed point problems from a unique common point of view. We have covered two of such auxiliary functions in Chapters 5 and 6. On the same direction, Khojasteh et al. [105] introduced the class of simulation functions which is also designed to unify several contractions in one go.

7.2

Simulation Function

Khojasteh et al. [105] introduced the notion of simulation function as follows (see Definition 1.2.9): A function ξ : [0, ∞) × [0, ∞) → R is said to be a simulation function if it satisfies the following conditions:

(ξ1) ξ (0, 0) = 0, (ξ2) ξ (y, x ) < x − y, for all x, y > 0, (ξ3) if {yn } and { xn } are sequences in (0, ∞) such that lim yn = lim xn > 0, then n→∞

n→∞

lim sup ξ (yn , xn ) < 0. n→∞

Argoubi et al. [24], slightly sharpened the idea of simulation functions by removing the condition (ξ1) while De Hierro et al. [156] modified the condition

(ξ3) by adding the hypothesis yn < xn , for all n. Therefore, in the sequel, by a simulation function we refer to a function ξ : [0, ∞) × [0, ∞) → R satisfying condition (ξ2) and the modified version of condition (ξ3) namely: if {yn } and { xn } are sequences in (0, ∞) such that lim yn = lim xn > 0 and yn < xn , for all n, then n→∞

n→∞

lim sup ξ (yn , xn ) < 0. n→∞

Here, we collect the following simulation functions. For more examples and

further work on this function, one can be referred to [24, 49, 67, 105, 126, 129, 156, 158]. Example 7.2.1. ξ (y, x ) = λx − y, for all x, y ∈ [0, ∞), where λ ∈ [0, 1). Example 7.2.2. ξ (y, x ) =

x x +1

− y, for all x, y ∈ [0, ∞).


109

Example 7.2.3. ξ (y, x ) = ψ( x ) − y, for all x, y ∈ [0, ∞), where ψ : [0, ∞) → [0, ∞) is a n non-decreasing function such that ∑∞ n=1 ψ ( t ) < ∞, for all t > 0.

Example 7.2.4. ξ (y, x ) = ψ( x ) − φ(y), for all x, y ∈ [0, ∞), where φ, ψ : [0, ∞) →

[0, ∞) are two continuous functions such that ψ(t) = φ(t) = 0 if and only if t = 0 and ψ(t) < t ≤ φ(t), for all t > 0. Example 7.2.5. ξ (y, x ) = x − ψ( x ) − y, for all x, y ∈ [0, ∞), where ψ : [0, ∞) → [0, ∞) is a lower semi-continuous function such that ψ(t) = 0 if and only if t = 0. Example 7.2.6. ξ (y, x ) = xψ( x ) − y, for all x, y ∈ [0, ∞), where ψ : [0, ∞) → [0, 1) is such that lim ψ(t) < 1, for all r > 0. t →r +

Referring to Definition 1.2.10, a mapping S : M → M is said to be Z -contraction with respect to a simulation function ξ if ξ (d(Su, Sv), d(u, v)) ≥ 0, ∀u, v ∈ M. Utilizing the notion of Z -contractions, Khojasteh et al. [105] proved the following result: Theorem 7.2.1. [105] Let ( M, d) be a complete metric space and S : M → M a Z contraction with respect to a simulation function ξ. Then S has a unique fixed point. From now on, α is a function on M × M to [0, ∞). As mentioned earlier, the idea of simulation function unifies several contractions of the existing literature. However, this idea, in its earlier form, can not deduce the following result which involve the function α. Theorem 7.2.2. [159] Let ( M, d) be a complete metric space and S : M → M a continuous mapping satisfying α(u, v)d(Su, Sv) ≤ ψ d(u, v) , for all u, v ∈ M, where ψ : [0, ∞) → n [0, ∞) is non-decreasing function such that ∑∞ n=1 ψ ( t ) < ∞, for all t > 0. Assume that

the following two conditions hold: (i) there exists u0 ∈ M such that α(u0 , Su0 ) ≥ 1, (ii) S is α-admissible, Then S has a fixed point.

110

C HAPTER 7: R ESULTS U NDER S IMULATION F UNCTION Further, the authors in [159, Theorem 2.3] gave the following sufficient condition

for the uniqueness of the fixed point (via Theorem 7.2.2): (iii) for all u, v ∈ M there exists z ∈ M such that α(u, z) ≥ 1 and α(v, z) ≥ 1. Recently, Karapınar [100] succeeded to deduce Theorem 7.2.2 via enlarging the class of simulation functions by embedding admissible function into the simulation one. To accomplish his objective, he utilized the following: Definition 7.2.1. [100] Let ( M, d) be a metric space. A mapping S : M → M is said to be an αZ -contraction with respect to a simulation function ξ if ξ (α(u, v)d(Su, Sv), d(u, v)) ≥ 0, ∀u, v ∈ M.

(7.1)

Notice that above definition coincides with Definition 1.2.10 for α ≡ 1. Remark 7.2.1. [100] If S is an αZ -contraction with respect to a simulation function ξ, then α(u, v)d(Su, Sv) < d(u, v), ∀u, v ∈ M.

(7.2)

Definition 7.2.2. [148] Let ( M, d) be a metric space. A mapping S : M → M is said to be a triangular α-orbital admissible if for all u, v ∈ M, (i) α(u, Su) ≥ 1 ⇒ α(Su, S2 u) ≥ 1, (ii) {α(u, v) ≥ 1 & α(v, Sv) ≥ 1} ⇒ α(u, Sv) ≥ 1. Theorem 7.2.3. [100] Let ( M, d) be a complete metric space and S : M → M an αZ contraction with respect to ξ. Suppose that (i) S is triangular α-orbital admissible, (ii) there exists u0 ∈ M such that α(u0 , Sx0 ) ≥ 1, (iii) either S is continuous (or if {un } is a sequence in M such that α(un , un+1 ) ≥ 1 for all n and un → u ∈ M as n → ∞, then there exists a subsequence {unk } of {un } such that α(unk , u) ≥ 1 for all k). Then S has a fixed point. Moreover, this fixed point is unique if α(u, v) ≥ 1 for all fixed points u and v of S.


111

Definition 7.2.3. [62] Let ( M, d) be a metric space. A subset E of M is said to be precomplete if every Cauchy sequence {un } in E converges to a point of M. In this chapter, we endeavor to prove Theorem 7.2.3 involving a pair of selfmappings.

7.3

Common Fixed Point Results

We begin with the following definitions which are needed in our subsequent discussion: Definition 7.3.1. A mapping S : M → M is said to be a triangular αT-admissible if for all u, v and w ∈ M, (i) α( Tu, Tv) ≥ 1 ⇒ α(Su, Sv) ≥ 1, (ii) {α( Tu, Tw) ≥ 1 and α( Tw, Tv) ≥ 1} ⇒ α( Tu, Tv) ≥ 1. Observe that, on setting T = I M , Definition 7.3.1 remains a stronger version of Definition 7.2.2. In this situation, S is said to be a triangular α-admissible (see [101]). Definition 7.3.2. A set M is said to be an αT-directed set if for every u, v ∈ M there exists w ∈ M such that α(u, Tw) ≥ 1 and α(v, Tw) ≥ 1. On choosing T = I M in Definition 7.3.2, M remains α-directed . Definition 7.3.3. A metric space ( M, d) is said to be an αT-regular if for every sequence

{ Tun } in M such that α( Tun , Tun+1 ) ≥ 1 for all n and { Tun } converges to some Tu ∈ T ( M) there exists a subsequence { Tunk } (of { Tun }) such that α( Tunk , Tu) ≥ 1 for all k. Moreover, if T = I M , M is said to be an α-regular. The following is a generalization of Definitions 1.2.10 and 7.2.1. Definition 7.3.4. Let ( M, d) be a metric space and S, T : M → M. Then S is said to be an αT-admissible Z -contraction with respect to a simulation function ξ if ξ (α( Tu, Tv)d(Su, Sv), d( Tu, Tv)) ≥ 0, ∀u, v ∈ M.

(7.3)

112


Remark 7.3.1. The mapping S in Definition 7.3.4 satisfies the following: α( Tu, Tv)d(Su, Sv) < d( Tu, Tv).

(7.4)

Now, we state and prove our first result as under: Theorem 7.3.1. Let ( M, d) be a metric space and S, T : M → M such that S is an αT-admissible Z -contraction with respect to a simulation function ξ. Suppose that the following conditions hold: (i) there exists u0 ∈ M such that α( Tu0 , Su0 ) ≥ 1, (ii) S( M) ⊆ T ( M), (iii) S is triangular αT-admissible, (iv) S is T-continuous, (v) S( M) is precomplete in T ( M), (vi) M is αT-directed set, (vii) the pair (S, T ) is weakly compatible . Then the pair (S, T ) has a common fixed point. Moreover, this common fixed point is unique if α( Tu, Tv) ≥ 1 for all u, v ∈ C (S, T ). Proof. Choose u0 as in (i). Firstly, we show the existence of a Cauchy sequence with the initial point Su0 . In view of (ii), we can define an increasing sequence { Tun } in S( M) such that Sun−1 = Tun for all n.

(7.5)

Now, α( Tu0 , Su0 ) ≥ 1 can be written as α( Tu0 , Tu1 ) ≥ 1 which (via (i) of Definition 7.3.1) implies α(Su0 , Su1 ) ≥ 1, i.e., α( Tu1 , Tu2 ) ≥ 1. Continuing this process inductively and using (ii) of Definition 7.3.1 , we find that (for all n, m with m > n ≥ 1), α( Tun , Tum ) ≥ 1.

(7.6)

In case Tum = Tum+1 , for some m ∈ N, then um is a coincidence point. Otherwise, d( Tun , Tun+1 ) > 0, for all n. Consequently,


113

0 ≤ ξ (α( Tun , Tun+1 )d(Sun , Sun+1 ), d( Tun , Tun+1 ))

= ξ (α( Tun , Tun+1 )d( Tun+1 , Tun+2 ), d( Tun , Tun+1 )) < d( Tun , Tun+1 ) − α( Tun , Tun+1 )d( Tun+1 , Tun+2 ),

(7.7)

which shows that {d( Tun , Tun+1 )} is a strictly decreasing sequence of positive real numbers having a limit r ≥ 0. If r 6= 0, then on letting n → ∞ on both sides of (7.7), we get lim α( Tun , Tun+1 ) = 1. Further, due to (ξ3), we have n→∞

0 ≤ lim sup ξ (α( Tun , Tun+1 )d( Tun+1 , Tun+2 ), d( Tun , Tun+1 )) < 0, n→∞

a contradiction. Therefore, lim d( Tun , Tun+1 ) = 0.

n→∞

(7.8)

Now, we prove that { Tun } is a bounded sequence. To establish the claim, assume on a contrary that { Tun } is unbounded. Then there exists a subsequence { Tunk } such that n1 = 1 and for each k ∈ N, nk+1 is the minimum positive integer such that d( Tunk , Tunk+1 ) > 1

(7.9)

d( Tunk , Tum ) ≤ 1, for nk ≤ m ≤ nk+1 − 1.

(7.10)

and

On using (7.9) and triangular inequality, we have 1 < d( Tunk , Tunk+1 )

≤ d( Tunk , Tunk+1 −1 ) + d( Tunk+1 −1 , Tunk+1 ) ≤ 1 + d( Tunk+1 −1 , Tunk+1 ), which on letting k → ∞ gives rise (in view of (7.8)) lim d( Tunk , Tunk+1 ) = 1.

k→∞

By (7.4), we have α( Tunk −1 , Tunk+1 −1 )d( Tunk , Tunk+1 ) < d( Tunk −1 , Tunk+1 −1 ).

114


On using (7.6), (7.9) and (7.10), we have 1 < α( Tunk −1 , Tunk+1 −1 )d( Tunk , Tunk+1 )

< d( Tunk −1 , Tunk+1 −1 ) ≤ d( Tunk −1 , Tunk ) + d( Tunk , Tunk+1 −1 ) ≤ d( Tunk −1 , Tunk ) + 1. On letting k → ∞ and using (7.8), we get lim d( Tunk −1 , Tunk+1 −1 ) = lim α( Tunk −1 , Tunk+1 −1 ) = 1.

k→∞

k→∞

Hence, on using (ξ3), we have 0 ≤ lim sup ξ (α( Tunk −1 , Tunk+1 −1 )d( Tunk , Tunk+1 ), d( Tunk −1 , Tunk+1 −1 )) < 0, n→∞

which is a contradiction. Therefore, { Tun } is a bounded sequence. Next, let cn := max{d( Tui , Tu j ) : i, j ≥ n}. Observe that {cn } is decreasing sequence of non-negative real numbers which is bounded due to the boundedness of { Tun }. Therefore, there exists some c ≥ 0 such that lim cn = c. If c 6= 0, then by n→∞

the definition of {cn }, for every k ∈ N there exist mk , nk such that mk > nk ≥ k with ck −

1 ≤ d( Tumk , Tunk ) ≤ ck k

so that lim d( Tumk , Tunk ) = c.

k→∞

(7.11)

Using (7.4), (7.6) and the triangular inequality, we have d( Tunk , Tumk ) ≤ α( Tunk −1 , Tumk −1 )d( Tunk , Tumk )

< d( Tunk −1 , Tumk −1 ) ≤ d( Tunk −1 , Tunk ) + d( Tunk , Tumk ) + d( Tumk , Tumk −1 ), which on letting k → ∞ (due to (7.8) and (7.11)), gives rise lim d( Tunk −1 , Tumk −1 ) = c and lim α( Tunk −1 , Tumk −1 ) = 1.

k→∞

k→∞

As S is αT-admissible Z contraction w.r.t. ξ, we have 0 ≤ lim sup ξ (α( Tunk −1 , Tumk −1 )d( Tunk , Tumk ), d( Tunk −1 , Tumk −1 )) < 0, n→∞

a contradiction. This shows that c = 0. Consequently, { Tun } is a Cauchy sequence


115

in S( M). The precompleteness of S( M) in T ( M) ensures the existence of some w ∈ M with lim Tun = Tw.

n→∞

(7.12)

Secondly, we show the existence of a common point of the pair (S, T ). If S is T-continues, then lim Sun = Sw which (in view of (7.5) and the uniqueness of the n→∞

limit) implies that Tw = Sw. Let z be such that z = Sw = Tw. In view of condition (vii), we have Sz = S( Tw) = T (Sw) = Tz. Suppose that d( Tz, z) > 0. Using (7.3) and (ξ3) 0 ≤ ξ (α( Tw, Tz)d(Sw, Sz), d( Tw, Tz) = ξ (α(z, Tz)d(z, Sz), d(z, Tz) < 0, a contradiction. Thus we have z = Sz = Tz. Hence, z is a common fixed point of the pair (S, T ). If z0 is another such point with d(z, z0 ) > 0, then 0 ≤ ξ (α( Tz, Tz0 )d(Sz, Sz0 ), d( Tz, Tz0 )

= ξ (α(z, z0 )d(z, z0 ), d(z, z0 ) < d(z, z0 ) − α(z, z0 )d(z, z0 ) ≤ 0, a contradiction so that the pair (S, T ) has a unique common fixed point. This concludes the proof. The following result is yet another version of Theorem 7.3.1. Theorem 7.3.2. Theorem 7.3.1 remains true if assumption (iv) is replaced by the following: (iv*) T ( M) is αT-regular. Proof. Firstly, on the lines of first step of the proof of Theorem 7.3.1, we can deduce a sequence { Tun } with initial point Su0 such that (7.12) holds. Secondly, the αTregularity of T ( M ) implies that there exists a subsequence { Tunk } of { Tun } with

116


α( Tunk , Tw) ≥ 1, for all k. Now, owing to (7.1), for all k, we have 0 ≤ ξ (α( Tunk , Tw)d(Sunk , Sw), d( Tunk , Tw))

= ξ (α( Tunk , Tw)d( Tunk +1 , Sw), d( Tunk , Tw)) < d( Tunk , Tw) − α( Tunk , Tw)d( Tunk +1 , Sw), which is equivalent to d( Tunk +1 , Sw) ≤ α( Tunk , Tw)d( Tunk +1 , Sw) < d( Tunk , Tw). Making k → ∞ one gets d( Tw, Sw) ≤ 0 implying thereby Sw = Tw. The rest of the proof can be completed on the lines of the proof of Theorem 7.3.1. Remark 7.3.2. The hypotheses of Theorems 7.3.1 and 7.3.2 up to assumption (iv) are enough to ensure the existence of the coincidence point of the underlying pair. Remark 7.3.3. The completeness assumption in our results is lightened by employing the precompleteness of the suitable range space. Remark 7.3.4. Observe that, asserting the α-directedness on the set of coincidence points of the underlying pair (of mappings) is relatively weaker than assuming it on the whole space. However, it remains an unnatural condition as computation of set of coincidence points is not always easy. Therefore, in our results, we use the directedness on whole space. Corollary 7.3.1. Let ( M, d) be a metric space and S : M → M such that S is α-admissible

Z -contraction with respect to a simulation function ξ. Suppose that the following conditions hold: (i) there exists u0 ∈ M such that α(u0 , Su0 ) ≥ 1, (ii) S is triangular α-admissible, (iii) either S is continuous or M is α-regular. (iv) ( M, d) is complete, (v) M is α-directed set, Then S has a fixed point. Moreover, this fixed point is unique if α(u, v) ≥ 1 for all u, v ∈ Fix (S).


117

To substantiate the advantage of our results, we derive the following two consequences. We begin with the following result which remains a generalized version of Theorem 7.2.2 involving two mappings: Corollary 7.3.2. Let ( M, d) be a metric space and S, T : M → M. Suppose that α( Tu, Tv)d(Su, Sv) ≤ ψ d( Tu, Tv) , ∀u, v ∈ M, where ψ is as in Theorem 7.2.2 . If conditions (i)-(vii) of Theorem 7.3.1 (resp. Theorem 7.3.2) are satisfied, then the pair (S, T ) has a unique common fixed point. Proof. The conclusion follows directly in view of Example 7.2.3 and Theorem 7.3.1 (resp. Theorem 7.3.2). Corollary 7.3.3. Let ( M, d) be a metric space and S : M → M. Suppose that α(u, v)d(Su, Sv) ≤ ψ d(u, v) , ∀u, v ∈ M,

(7.13)

where ψ is as in Theorem 7.2.2 . If conditions (i)-(v) of Corollary 7.3.1 are satisfied, then S has a unique fixed point. Now, we employ our proved result to deduce the following order-theoretic fixed point results Corollary 7.3.4. Theorems 1.3.1 and 1.3.2 follow immediately from Corollary 7.3.3. We prove Corollary 7.3.4 concerning Theorem 1.3.1. The proof for Theorem 1.3.2 is very similar and hence omitted. Proof. Define a mapping α : M × M → [0, ∞) by   1, f or u v α(u, v) =  0, otherwise. From condition (v) of Theorem 1.3.1, one can see that S satisfies (7.13) for ψ(t) = λt for λ ∈ [0, 1). Now, let u, v ∈ M such that α(u, v) ≥ 1. By the definition of α, we have u v. Since S is an increasing mapping, we have Su Sv so that that α(Su, Sv) = 1. Then S is α-admissible. From condition (iv) of Theorem 1.3.1, there exists u0 ∈ M such that u0 Su0 so that α(u0 , Su0 ) = 1. Moreover, if α(u, w) ≥ 1

118


and α(w, v) ≥ 1 (i.e., u w and w v) then via the transitivity, u v and hence α(u, v) = 1. Therefore, all the hypotheses of Corollary 7.3.3 are satisfied so that S has a fixed point. The following result appears to be new in the existing literature which is essentially inspired by Example 7.2.4. Corollary 7.3.5. Let ( M, d) be a metric space and S, T : M → M. Suppose that φ α( Tu, Tv)d(Su, Sv) ≤ ψ d( Tu, Tv) , ∀u, v ∈ M, where φ, ψ : [0, ∞) → [0, ∞) are two continuous mappings such that ψ(t) = φ(t) = 0 ⇔ t = 0 and ψ(t) < t ≤ φ(t), for all t > 0. If conditions (i)-(vii) of Theorem 7.3.1 (resp. Theorem 7.3.2) are satisfied, then the pair (S, T ) has a unique common fixed point. Similarly, common fixed point theorems can also be derived corresponding to various simulation functions available in the existing literature.

7.4

An Application

Inspired by [159], we establish the existence solution for the following two-point boundary value problem of second order differential equation:   u00 (t) = f (t, u(t)), t ∈ J = [0, 1]

(7.14)

 u(0) = u(1) = 0, where f : J × R → R is a continuous function. The Green function G (t, s) associated to (7.14) is given by G (t, s) =

   t (1 − s ),

0 ≤ t ≤ s ≤ 1,

  s (1 − t ),

0 ≤ s ≤ t ≤ 1.

.

Let C( J ) denotes the space of all continuous functions defined on J. We know that (C( J ), d) is a complete metric space where d(u, v) = sup |u(t) − v(t)|e−t . t∈ J

(7.15)


119

Consider the following conditions: (a) There exists a function ω : R2 → R such that for all t ∈ J and for all a, b ∈ R with ω ( a, b) ≥ 0,

| f (t, a) − f (t, b)| ≤ ψ(|b − a|e−t ) where ψ is as in Theorem 7.2.2. (b) There exists u0 ∈ C( J ) such that for all t ∈ J, we have Z 1 ω u0 ( t ), G (t, s) f (s, u0 (s))ds ≥ 0. 0

(c) For all t ∈ J and all u, v ∈ C( J ) Z Z 1 ω (u(t), v(t)) ≥ 0 ⇒ ω G (t, s) f (s, u(s))ds, 0

1 0

G (t, s) f (s, v(s))ds ≥ 0.

(d) If un → u ∈ C( J ) and ω (un+1 , un ) ≥ 0, for all n ∈ N, then ω (un , u) ≥ 0 for all n ∈ N. Now, we prove the following result on the existence and uniqueness of a solution of the problem described by (7.14). Theorem 7.4.1. Problem (7.14) has at least one solution u∗ ∈ C 2 provided the conditions (a)-(d) hold. Proof. Notice that an element u ∈ C 2 is a solution of the problem described by (7.14) if u ∈ C is a solution of the integral equation u(t) =

Z 1 0

G (t, s) f (s, u(s))ds,

∀t ∈ J.

(7.16)

∀t ∈ J.

(7.17)

Define a function S : C( J ) → C( J ) by Z 1

Su(t) =

0

G (t, s) f (s, u(s))ds,

Clearly, if u ∈ C( J ) is a fixed point of S, then u ∈ C( J ) is a solution of (7.16) and hence of (7.14). Let u, v ∈ C( J ) such that ω (u(t), v(t)) ≥ 0, for all t ∈ J, then by (a), we have Z 1 Z 1 |Su(t) − Sv(t)| = G (t, s) f (s, u(s))ds − G (t, s) f (s, v(s))ds 0

≤

Z 1 0

≤ 8

0

G (t, s)ψ(|v(s) − u(s)|e−s )ds

Z 1 0

G (t, s)ψ(|v(s) − u(s)|e−s )ds

120


≤ 8 sup t∈ J

Z 1 0

= ψ(d(u, v)),

! G (t, s)ds ψ(d(u, v)) (as sup t∈ J

Z 1 0

G (t, s)ds =

1 ). 8

Thus, |Su(t) − Sv(t)|e−t ≤ ψ(d(u, v)) so that d(Su, Sv) ≤ ψ(d(u, v)) ∀u, v ∈ C( J ) with ω (u(t), v(t)) ≥ 0. Define a function α : C( J ) × C( J ) → [0, ∞) by:   1 if ω (u(t), v(t)) ≥ 0, t ∈ J α(u, v) = .  0 otherwise. Then for all u, v ∈ C( J ), we have α(u, v)d(Su, Sv) ≤ ψ(d(u, v)), i.e., condition (7.13) is satisfied. Further, condition (c) implies that α(u, v) ≥ 1 ⇒ ω (u(t), v(t)) ⇒ ω (Su(t), Sv(t)) ≥ 0 ⇒ α(Su, Sv) ≥ 1. So that S is α-admissible mapping. Also, from (b), there exists u0 ∈ C( J ) such that for all t ∈ J, we have α(u0 (t), Su0 ) ≥ 1. Finally, (d) and Corollary 7.3.3 ensures the existence of some u∗ ∈ C such that u∗ = Su∗ i.e, u∗ is a solution of (7.14).

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