Metric Spaces - ARC Journals

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 3, Issue 9, September 2015, PP 35-48 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org

Some New Fixed Point and Common Fixed Point Theorems in Dislocated Metric Spaces and Dislocated Quasi -Metric Spaces Manoj Ughade1, R. D. Daheriya2 1 2

Department of Mathematics, Sarvepalli Radhakrishnan University, Bhopal, Madhya Pradesh, India

Department of Mathematics, Government J.H. Post Graduate College, Betul, Madhya Pradesh, India

Abstract: Our goal in this paper is to study of fixed point theorems in dislocated quasi-metric spaces and dislocated metric spaces which we can apply in a variety of different situations. This article can be considered as a continuation of the remarkable works of Hitzler et al [9], Zeyada et al. [6] and Geraghty [17]. In this article, we review briefly some generalizations of metric space with examples and we describe some properties, introduce new definitions and present some lemmas and propositions related to dislocated metric spaces and dislocated quasi-metric spaces. We also present some fixed point and common fixed point theorems for selfmappings in a complete dislocated metric spaces and quasi-metric spaces under various contractive conditions and present some examples to illustrate the effectiveness of our results. Keywords and Phrases: Fixed point, Common fixed point, dislocated metric spaces, dislocated quasi-metric spaces, contraction. MATHEMATICS SUBJECT CLASSIFICATION (2010): 47H10, 54H25;

1. INTRODUCTION AND PRELIMINARES Fixed point theory is one of the most dynamic research subjects in nonlinear sciences. Regarding the feasibility of application of it to the various disciplines, a number of authors have contributed to this theory with a number of publications. One of the simplest and most useful results in fixed point theory is the Banach fixed point theorem: Let ሺܺ, ݀ሻbe a complete metric space and T be self mapping of X satisfying ݀ሺܶ‫ݔ‬, ܶ‫ݕ‬ሻ ≤ ߣ݀ሺ‫ݔ‬, ‫ݕ‬ሻ for all ‫ݔ‬, ‫ܺ ∈ ݕ‬,

where ߣ ∈ ሾ0,1ሻ, then T has a unique fixed point. A mapping satisfying the condition (1.1) is called contraction mapping. As well as, there are a lot of extensions of this famous fixed point theorem in metric space which are obtained generalizing contractive condition, there are a lot of generalizations of it in different space which has metric type structure. In fact, Banach demonstrated how to find the desired fixed point by offering a smart and plain technique. This elementary technique leads to increasing of the possibility of solving various problems in different research fields. This celebrated result has been generalized in many abstract spaces for distinct operators. (1.1)

The concept of dislocated metric space was introduced by P. Hitzler [3] in which the self distance of points need not to be zero necessarily. They also generalized famous Banach’s contraction principle in dislocated metric space. Dislocated metric space play a vital rule in topology, logical programming and electronic engineering. F. M. Zeyada et al.[6] develops the notation of complete dislocated quasi metric spaces and generalized the result of Hitzler [3] in dislocated quasi metric space. In [2] C.T. Aage and J. N. Salunke proved dislocated quasi-metric version of Kannan mapping theorem. AminiHaradi [7] re-introduced the dislocated space under the name of a metric-like space and proved some fixed theorems in this space. Very recently, Bennani et al. [8] established two new common fixed point theorems for four self maps on dislocated metric spaces, which improved the results of Panthi and Jha [10] without any continuity requirement. After F. M. Zeyada et al.[6] many papers have been published containing fixed point results in dislocated quasi metric spaces (see [1, 2, 4, 5, 11,12,13,14, 29, 30, 31]).

©ARC

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Manoj Ughade & R. D. Daheriya

Throughout this paper ℕ denotes the set of natural numbers, ℕ଴ = ℕ ∪ ሼ0ሽ, ℝ denotes the set of all real numbers, ℝା = ሺ0, ∞ሻ, ℝା ଴ = ሾ0, ∞ሻ. Now, we will discuss difference between metric space and generalizations of metric space (quasi-metric, pseudo-metric, dislocated metric, dislocated quasimetric, partial metric, ultra metric). The following definition will be needed in the sequel. Definition 1.1 Consider a non-empty set X, whose elements will be refered to as points. A distance function on a set X is a function d: X × X → ℝା ଴ which assigns to each pair of points x and y in X a real number dሺx, yሻ.We need the following conditions: (1) Non-negativity or Separation Axiom: ∀ x, y ∈ X, dሺx, yሻ ≥ 0;

(2) Identity of Indiscernible or Faithful : ∀ x, y ∈ X, dሺx, yሻ = 0 ⇔ x = y;

(3) Small self-distance: ∀ x ∈ X, dሺx, xሻ = 0; (But possibly dሺx, yሻ = 0 for some distinct values x ≠ y). (4) Indistancy Implies Equality: ∀ x, y ∈ X, dሺx, yሻ = 0 = dሺy, xሻ ⇒ x = y;

(5) Equality : ∀ x, y ∈ X, x = y ⇔ dሺx, xሻ = dሺx, yሻ = dሺx, yሻ ; (6) Small self-distances: ∀ x, y ∈ X, dሺx, xሻ ≤ dሺx, yሻ;

(7) Symmetry: ∀ x, y ∈ X, dሺx, yሻ = dሺy, xሻ;

(8) The Triangle Inequality: ∀ x, y, z ∈ X, dሺx, yሻ ≤ dሺx, zሻ + dሺz, yሻ;

(9) The Triangle Inequality: ∀ x, y, z ∈ X, dሺx, yሻ ≤ dሺx, zሻ + dሺz, yሻ − dሺz, zሻ; (10) Strong Triangle Inequality: ∀ x, y, z ∈ X, dሺx, yሻ ≤ maxሼdሺx, zሻ, dሺz, yሻሽ;

If d satisfies conditions (1), (2), (7) and (8), then it is called a metric on X. If it satisfies conditions (1), (2), and (8), it is called a quasi-metric on X. If a metric d satisfies the strong triangle inequality (10), then it is called an ultra metric. If it satisfies conditions (1), (5), (6) and (9), it is called a partial metric. If it satisfies conditions (1), (4), (7) and (8), it is called a dislocated metric (or simply dmetric) on X and the pair ሺX, dሻ is called a dislocated metric space. Moreover "dሺx, yሻ = 0 ⇒ x = y" when d is a d-metric. However "x = y" does not necessarily imply "dሺx, yሻ = 0" when d is a d-metric. If d is a d-metric instead of a metric, it is possible that dሺx, xሻ ≠ 0.As such these implications hold well in a d-metric space as well when "x ≠ y" is replaced by dሺx, yሻ ≠ 0. If it satisfies conditions (1), (4), and (8), it is called a dislocated quasi-metric (or simply d୯ -metric).

Clearly every metric space is a dislocated metric space but the converse is not necessarily true as clear form the following example. Example 1.2 Let X = ሾ0,1] define the distance function d ∶ X × X → ℝା by dሺx, yሻ = maxሼx, yሽ. Clearly X is dislocated metric space but not a metric space.

Also every metric space is dislocated quasi metric space but the converse is not true and every dislocated metric space is dislocated quasi metric space but the converse is not true as clear from the following example. Example 1.3 Let X = ሾ0, 1], we define the function d ∶ X × X → ℝା as dሺx, yሻ = |x − y| + |x| for all x, y ∈ X. Clearly X is dq-metric space but not a metric space nor dislocated metric space.

Definition 1.4 A sequence ሺx୬ ሻ in d୯ -metric space ሺX, dሻ with respect to d୯ is said to be d୯ converge to x ∈ X provided that In our main work we will use the following definitions which can be found in [3, 6]. d୯ − lim୬→∞ dሺx୬ , xሻ = d୯ − lim୬→∞ dሺx, x୬ ሻ = 0 .

In this case x is called the d୯ −limit of ሺx୬ ሻ and we write d୯ − lim୬→∞ x୬ = x or x୬ → x as n → ∞.

Definition 1.5 A sequence ሺx୬ ሻ in d −metric space ሺX, dሻ with respect to d is said to be d –converge to x ∈ X provided that International Journal of Scientific and Innovative Mathematical Research (IJSIMR)

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Some New Fixed Point Theorems and Common Fixed Point Theorems in Dislocated Metric Spaces and Dislocated Quasi -Metric Spaces

d − lim୬→∞ dሺx୬ , xሻ = d − lim୬→∞ dሺx, x୬ ሻ = 0 .

In this case x is called the d −limit of ሺx୬ ሻ and we write d − lim୬→∞ x୬ = x or x୬ → x as n → ∞.

Definition 1.6 We call a sequence ሺx୬ ሻ in d୯ -metric space (d-metric space) ሺX, dሻ is a d୯ -Cauchy ( d -Cauchy) sequence provided that for all ϵ > 0 there exists n଴ = n଴ ሺϵሻ ∈ ℕ such that dሺx୬ , x୫ ሻ < ϵ, ∀ m, n ≥ n଴ .

Definition 1.7 A d୯ -metric space (d-metric space) ሺX, dሻ is called d୯ -complete (d-complete) if every d୯ -Cauchy (d-Cauchy) sequence in X converges with respect to d in X. We discuss a few more examples of quasi-metric spaces, d-metrics and d୯ -metrics.

Example 1.8 Let X be the set of positive integers and define the non-negative real valued function d: X × X → ℝା ଴ by dሺx, yሻ = ൞

ଵ ୷

,

if x < ‫ݕ‬

0, 1,

if x = y if x > ‫ݕ‬

Note that d satisfies all the axioms of quasi-metric and therefore the pair ሺX, dሻ is a quasi-metric space but it is not a metric space because d is non-symmetric, i.e. dሺx, yሻ ≠ dሺy, xሻ. Example 1.9 Let X = ℝ. Define d: X × X → ℝା ଴ by dሺx, yሻ = ൜

0, |y|

if x = y otherwise,

Then one can easily see that d is a quasi-metric and the pair ሺX, dሻ is a quasi-metric space.

Example 1.10 Let X = ሾ0,1]. Define d: X × X → ℝା ଴ by 1. dሺx, yሻ = ቊ

2.

dሺx, yሻ = ൝

y−x,

2ሺx − yሻ,

x−y,

ଵ ሺy − ଶ

xሻ,

if x ≤ y,

otherwise, if x ≥ y,

otherwise,

Then here also d is a quasi-metric and the pair ሺX, dሻ is a quasi-metric space.

ଶ Example 1.11 Define d: ℝ × ℝ → ℝା ଴ by dሺx, yሻ = |x| + y , then d is d୯ - metric on ℝ which is not a d - metric and the pair ሺℝ, dሻ is a d୯ - metric space. ା Example 1.12 Let X = ℝା ଴ and define a distance function d: X × X → ℝ଴ such that dሺx, yሻ = maxሼx, yሽ. Then d is a d-metric on X and the pair ሺX, dሻ is a d-metric space.

Example 1.13 Let X = ൜ଶ౤ | n ∈ ℕ⋃ሼ0ሽൠ. Define d: X × X → Rା଴ by setting dሺx, yሻ = ሺx − yሻଶ for ଵ

all x, y ∈ X. Then d is a d-metric on X and the pair ሺX, dሻ is a d-metric space.

ା Example 1.14 Let X = ℝା ଴ . Then d: X × X → ℝ଴ define by dሺx, yሻ = ଶ |x − y| + ଶ ሺx + yሻ. for all x, y ∈ Rା଴ is a d-metric on Rା଴ and the pair ሺX, dሻ is a d-metric space. ଵ

ଵ

Example 1.15 Let I be the set of all closed interval on ℝ . Then d: I × I → ℝା ଴ defined ሾc, by dሺሾa, b], d]ሻ = maxሼb, dሽ − minሼa, cሽ for all ሾa, b], ሾc, d] ∈ I, is a d-metric on X and ሺI, dሻ is a d-metric space. Example 1.16 Let X = ሼ0,1,2ሽ and a mapping d: X × X → ℝା ଴ be defined as dሺ0,0ሻ = dሺ1,1ሻ = dሺ2,2ሻ = 0,

dሺ0,1ሻ = dሺ1,2ሻ = dሺ0,2ሻ = 1, dሺ1,0ሻ = dሺ2,1ሻ = dሺ2,0ሻ = 2.

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Note that d satisfies all the axioms of d୯ -metric and the pair ሺX, dሻ is a d୯ -metric space but it is not a d-metric space because d is non-symmetric i.e. dሺx, yሻ ≠ dሺy, xሻ.

ା Example 1.17 Let X = ℝା ଴ and define the function d: X × X → ℝ଴ by dሺx, yሻ = |x − y| + |x| for all x, y ∈ X evidently d is d୯ -metric but not a metric nor d-metric.

Example 1.18 Let X = ሼ0,1ሽ and a mapping d: X × X → ℝା ଴ be defined as dሺ1,1ሻ = dሺ2,2ሻ = 0, dሺ1,2ሻ = dሺ2,1ሻ = 1. We observe that d satisfies all the axioms of d-metric and the pair ሺX, dሻ is a d-metric space. Also ሺX, dሻ is a d୯ -metric space. 1) any point ‫ ܺ ∈ ݔ‬is said to be fixed point of ܶ if ܶ‫;ݔ = ݔ‬

Definition 1.19 Let S and T be two self-mappings on a nonempty set X; then

2) any point ‫ ܺ ∈ ݔ‬is called coincidence point of ܵ and ܶ if ܵ‫ ݔܶ = ݔ‬and one calls ‫ݔܶ = ݔܵ = ݑ‬ a point of coincidence of ܵ and ܶ; 3) a point ‫ ܺ ∈ ݔ‬is called common fixed point of ܵ and ܶ if ܵ‫ݔ = ݔܶ = ݔ‬.

Lemma 1.20 (see [6]) Let ሺX, dሻ be a dq-metric space. If ݂ ∶ ܺ → ܺ is a contraction function, then ݂ ௡ ሺ‫ݔ‬଴ ሻ is a Cauchy sequence for each x଴ ∈ X .

Lemma 1.21 (see [6]) Every subsequence of d୯ -convergent sequence to a point x଴ is d୯ - convergent to x଴ . Theorem 1.22 (see [6]) Let ሺX, dሻ be a d୯ -metric space and let ܶ: ܺ → ܺ be continuous contraction mapping. Then ܶ has a unique fixed point.

Theorem 1.23 (see [19]) Let ሺX, dሻ be a complete d-metric space and let ܶ: ܺ → ܺ be a contraction. Then ܶ has a unique fixed point.

2. MAIN RESULT

In this section, first we explore the properties of dislocated metric space and dislocated quasi-metric space. Proof Let ሺx୬ ሻ be a sequence which converges to some x, and let ϵ > 0 be arbitrarily chosen. Then ஫ there exists n଴ ∈ ℕ with dሺx୬ , xሻ < ଶ for all n ≥ n଴ . For m, n ≥ n଴ , we then obtain dሺx୫ , x୬ ሻ ≤ Proposition 2.1 Every converging sequence in a d-metric space is a Cauchy sequence. dሺx୫ , xሻ + dሺx, x୬ ሻ < ଶ + ଶ = ϵ. Hence ሺx୬ ሻ is a Cauchy sequence. ஫

஫

Proof Let x ∗ and y ∗ be limits of the sequence ሼx୬ ሽ .Then dሺx୬ , x ∗ ሻ → 0 and dሺx୬ , y ∗ ሻ → 0 as n → ∞. By the triangle inequality, it follows that dሺx ∗ , y ∗ ሻ ≤ dሺx ∗ , x୬ ሻ + dሺx୬ , y ∗ ሻ = dሺx୬ , x ∗ ሻ + dሺx୬ , y ∗ ሻ → 0 as n → ∞. Hence dሺx ∗ , y ∗ ሻ = 0 and so x ∗ = y ∗ . Proposition 2.2 Limits in d-metric spaces are unique.

Proposition 2.3 Limits in d୯ -metric spaces are unique.

Proof Let x ∗ and y ∗ be limits of the sequence ሺx୬ ሻ .Then x୬ → x ∗ and x୬ → y ∗ as n → ∞. By the triangle inequality, it follows that dሺx ∗ , y ∗ ሻ ≤ dሺx ∗ , x୬ ሻ + dሺx୬ , y ∗ ሻ . Letting n → ∞ , we obtain dሺx ∗ , y ∗ ሻ ≤ dሺx ∗ , x ∗ ሻ + dሺy ∗ , y ∗ ሻ. Similarly dሺy ∗ , x ∗ ሻ ≤ dሺx ∗ , x ∗ ሻ + dሺy ∗ , y ∗ ሻ . Hence |dሺx ∗ , y ∗ ሻ − dሺy ∗ , x ∗ ሻ| ≤ 0 and so dሺx ∗ , y ∗ ሻ = dሺy ∗ , x ∗ ሻ . Also dሺx ∗ , y ∗ ሻ ≤ dሺx ∗ , x୬ ሻ + dሺx୬ , y ∗ ሻ → 0 as n → ∞ and therefore dሺx ∗ , y ∗ ሻ = dሺy ∗ , x ∗ ሻ = 0. It follows that x ∗ = y ∗ . Definition 2.4 Let ሺX, dሻ is a d-metric space. Given a point x଴ ∈ X and a real number ϵ > 0. We define three types of sets. Now, we introduce the following.

Open Ball: Closed Ball: Sphere:

ℬ஫ୢ ሺx଴ ሻ = ሼ x ∈ X | dሺx଴ , xሻ < ϵ ሽ

෩஫ୢ ሺx଴ ሻ = ሼ x ∈ X | dሺx଴ , xሻ ≤ ϵ ሽ ℬ

࣭஫ୢ ሺx଴ ሻ = ሼ x ∈ X | dሺx଴ , xሻ = ϵ ሽ


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In all the above three cases x଴ is called the centre and ϵ the radius. We observe that ෩஫ୢ ሺx଴ ሻ − ℬ஫ୢ ሺx଴ ሻ. ࣭஫ୢ ሺx଴ ሻ = ℬ

If ሺX, dሻ is a d୯ -metric space. Given a point x଴ ∈ X and a real number ϵ > 0. Then Open Ball:

Closed Ball: Sphere:

ℬ஫ ౧ ሺx଴ ሻ = ሼ x ∈ X | minሼdሺx଴ , xሻ, dሺx, x଴ ሻሽ < ϵ ሽ ୢ

෩஫ୢ౧ ሺx଴ ሻ = ሼ x ∈ X | minሼdሺx଴ , xሻ, dሺx, x଴ ሻሽ ≤ ϵ ሽ ℬ

࣭஫ ౧ ሺx଴ ሻ = ሼ x ∈ X | minሼdሺx଴ , xሻ, dሺx, x଴ ሻሽ = ϵ ሽ ୢ

In all the above three cases x଴ is called the centre and ϵ the radius. Note that ୢ ෩஫ୢ౧ ሺx଴ ሻ − ℬ஫ୢ౧ ሺx଴ ሻ. ࣭஫ ౧ ሺx଴ ሻ = ℬ

There is no guarantee that x ∈ ℬ஫ ౧ ሺxሻ, for example, in example 1.14, if x > 0 and y > 0, then 1 ∉ ℬభ ౧ . ୢ

ୢ

ర

Definition 2.5 Neighbourhood is an (open ϵ −) ball in a d-metric space ሺX, dሻ with center x ∈ X is a set ࣨ஫ୢ ሺxሻ = ሼy ∈ X | dሺx, yሻ < ϵሽ where ϵ > 0 .

Note that an (open ϵ −) ball may be empty in d-metric space. In fact, the centre of an (open ϵ −) ball is contained in the (open ϵ −) ball itself; the point may be dislocated from the ball.

In a d୯ -metric space ሺX, dሻ , a neighbourhood is an (open ϵ − ) ball with center x ∈ X is a set ࣨ஫ ౧ ሺxሻ = ሼy ∈ X | minሼdሺx, yሻ, dሺy, xሻሽ < ϵሽ where ϵ > 0. ୢ

Now, we give the following two Propositions.

Proposition 2.6 Let ሺX, dሻ be a dislocated metric space. Then for all x ∈ X. dሺx, xሻ = 0 iff ࣨ஫ୢ ሺxሻ ≠ ∅. for all ϵ > 0 and x ∈ X.

Proof Since ࣨ஫ୢ ሺxሻ = ሼy ∈ X | dሺx, yሻ < ϵሽ, where ϵ > 0, is an open ϵ −ball in a d-metric space ሺX, dሻ with center x ∈ X. Suppose small self distance: dሺx, xሻ = 0 for all x ∈ X and ϵ > 0 be given, then dሺx, xሻ < ϵ ⇒ x ∈ ࣨ஫ୢ ሺxሻ . Hence ࣨ஫ୢ ሺxሻ ≠ ∅ for all and x ∈ X . Conversely, suppose that ࣨ஫ୢ ሺxሻ ≠ ∅ for all ϵ > 0. We know that ࣨ஫ୢ ሺxሻ = ሼy ∈ X | dሺx, yሻ < ϵሽ . Then ∃ y ∈ X such that dሺx, yሻ = r < ϵ. By using triangle inequality and symmetry property, it follows that dሺx, xሻ ≤ dሺx, yሻ + dሺy, xሻ = dሺx, yሻ + dሺx, yሻ = 2dሺx, yሻ = 2r and hence for all ϵ > 0, ݀ሺx, xሻ < ϵ . Therefore dሺx, xሻ = 0. Proposition 2.7 Let ሺX, dሻ be a d-metric space. Then d is a metric if and only if for all x ∈ X and all ε > 0, ࣨ஫ୢ ሺxሻ ≠ ∅.

Proof Let d is a metric. We then have dሺx, xሻ = 0 ∀ x ∈ X . By Proposition ሺ2.6ሻ , it follows that ࣨ஫ୢ ሺxሻ ≠ ∅ . Conversely, let ࣨ஫ୢ ሺxሻ ≠ ∅ for all x ∈ X and all ε > 0. Since ࣨ஫ୢ ሺxሻ = ሼy ∈ X|dሺx, yሻ < ϵሽ. Then dሺx, xሻ < ϵ. Hence for all x ∈ X, dሺx, xሻ = 0, and then dሺx, yሻ = 0 if and only if x = y. Also symmetric and triangular properties hold. Hence d is a metric.

Definition 2.8 Let X = ሺX, dሻ and Y = ሺY, d෨ሻ be two metric spaces. A mapping f: X → Y is said to be continuous at a point x଴ ∈ X, if for every ϵ > 0, there is a δ > 0 such that d෨ሺfx, fx଴ ሻ < ϵ for all x satisfying dሺx, x଴ ሻ < δ. Mapping f is said to be continuous, if it is continuous at every point of X. Theorem 2.9 A mapping ݂: ܺ → ܻ of a d-metric space ሺX, dሻ into a d-metric space ሺY, d෨ሻ is continuous at a point ‫ݔ‬଴ ∈ ܺ if and only if ‫ݔ‬௡ → ‫ݔ‬଴ implies that ݂‫ݔ‬௡ → ݂‫ݔ‬଴ .

Proof Assume ݂ to be continuous at x଴ . Then for given ϵ > 0, there is a δ > 0 such that dሺx, x଴ ሻ < δ implies d෨ሺfx, fx଴ ሻ < ϵ. Let ሺx୬ ሻ be a sequence in X such that x୬ → x଴ . Then there is a n଴ such that for all n > n଴ , we have dሺx୬ , x଴ ሻ < δ. Hence for all n > n଴ , d෨ሺfx୬ , fx଴ ሻ < ϵ. By definition this means that fx୬ → fx଴ . Conversely we assume that T is continuous at x଴ . suppose this is false. Then there is an ϵ > 0 such that for every δ > 0, there is an x ≠ x଴ satisfying dሺx, x଴ ሻ < δ but d෨ሺfx, fx଴ ሻ ≥ ϵ . International Journal of Scientific and Innovative Mathematical Research (IJSIMR)

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In particular for δ = ୬ , there is an x୬ satisfying dሺx୬ , x଴ ሻ < ୬ , but d෨ሺfx୬ , fx଴ ሻ ≥ ϵ. Clearly x୬ → x଴ but ሺfx୬ ሻ does not converge to fx଴ . This contradicts fx୬ → fx଴ and proves the theorem. ଵ

ଵ

ିଵ Theorem 2.10 Let τ: ℝା ଴ → ሾ0, 2 ሻ be decreasing function. Let ሺX, dሻ be a complete d-metric space and let T: X → X be a self-mapping such that

Now, we are ready to state and prove our first main results.

dሺTx, Tyሻ ≤ τ൫dሺx, yሻ൯൫dሺx, Txሻ + dሺy, Tyሻ൯

(2.1)

for all x, y ∈ X. If moreover, T is continuous or τ is a constant function then T has a unique fixed point x ∗ ∈ X.

Proof Let x଴ be an arbitrary point in X and we set x୬ାଵ = Tx୬ for each n ∈ ℕ଴ . We may assume that dሺx୬ , x୬ାଵ ሻ ≠ 0 for each n ∈ ℕ଴ . Then by (2.1), we get dሺx୬ାଵ , x୬ାଶ ሻ = dሺTx୬ , Tx୬ାଵ ሻ

≤ τ൫dሺx୬ , x୬ାଵ ሻ൯൫dሺx୬ , Tx୬ ሻ + dሺx୬ାଵ , Tx୬ାଵ ሻ൯

(2.2)

= τ൫dሺx୬ , x୬ାଵ ሻ൯൫dሺx୬ , x୬ାଵ ሻ + dሺx୬ାଵ , x୬ାଶ ሻ൯ ≤ 2ିଵ ൫dሺx୬ , x୬ାଵ ሻ + dሺx୬ାଵ , x୬ାଶ ሻ൯

Hence, dሺx୬ାଵ , x୬ାଶ ሻ ≤ dሺx୬ , x୬ାଵ ሻ for each n ∈ ℕ଴ . So ሼdሺx୬ , x୬ାଵ ሻሽ is a nonnegative decreasing sequence. Hence, there exists γ ≥ 0 such that lim୬→ஶ dሺx୬ x୬ାଵ ሻ = γ. We claim that γ = 0. Suppose, on the contrary, that γ > 0. Then due to (2.2), we have dሺx୬ାଵ , x୬ାଶ ሻ ≤ τሺγሻ൫dሺx୬ , Tx୬ ሻ + dሺx୬ାଵ , Tx୬ାଵ ሻ൯ = τሺγሻ൫dሺx୬ , x୬ାଵ ሻ + dሺx୬ାଵ , x୬ାଶ ሻ൯

(2.3)

and consequently γ ≤ 2τሺγሻγ, which is impossible since τሺγሻ < 2ିଵ . This proves that γ = 0. Now we will show that ሺx୬ ሻ is a Cauchy sequence. From (2.1), we get dሺx୬ , x୫ ሻ = dሺfx୬ , fx୫ ሻ

≤ ଶ ሾdሺx୬ , fx୬ ሻ + dሺx୫ , fx୫ ሻ] ଵ

(2.4)

= ଶ ሾdሺx୬ , x୬ାଵ ሻ + dሺx୫ , x୫ାଵ ሻ] ଵ

There exist an ݊଴ such that for ݉, ݊ ≥ ݊଴, ݀ሺ‫ݔ‬௡ , ‫ݔ‬௡ାଵ ሻ < ߳ and ݀ሺ‫ݔ‬௠ , ‫ݔ‬௠ାଵ ሻ < ߳ . Hence ݀ሺ‫ݔ‬௡ , ‫ݔ‬௠ ሻ ≤ ଶ ሺ߳ + ߳ሻ = ߳ ଵ

(2.5)

for all ݉, ݊ ≥ ݊଴. This forces that ሺ‫ݔ‬௡ ሻ is Cauchy sequence and view of completeness of X, there exists an ‫ ܺ ∈ ∗ ݔ‬such that ‫ݔ‬௡ → ‫ ∗ ݔ‬. We can check that ܶ‫ ∗ ݔ = ∗ ݔ‬. If T is continuous, then ܶ‫ܶ = ∗ ݔ‬ሺ݈݅݉௡→ஶ ‫ݔ‬௡ ሻ = ݈݅݉௡→ஶ ܶ‫ݔ‬௡ = ݈݅݉௡→ஶ ‫ݔ‬௡ାଵ = ‫∗ ݔ‬

and ‫ ∗ ݔ‬is a fixed point of T. On the other hand, if ߬ is a constant. Then, we have ݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ ≤ ݀ሺ‫ ∗ ݔ‬, ‫ݔ‬௡ାଵ ሻ + ݀ሺ‫ݔ‬௡ାଵ , ܶ‫ ∗ ݔ‬ሻ

Letting ݊ → ∞, we get Similarly,

= ݀ሺ‫ ∗ ݔ‬, ‫ݔ‬௡ାଵ ሻ + ݀ሺܶ‫ݔ‬௡ , ܶ‫ ∗ ݔ‬ሻ

(2.6)

(2.7)

≤ ݀ሺ‫ ∗ ݔ‬, ‫ݔ‬௡ାଵ ሻ + ߬ ൫݀ሺ‫ݔ‬௡ , ‫ݔ‬௡ାଵ ሻ + ݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ൯

݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ ≤ ߬ ݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ.

݀ሺܶ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ ≤ ݀ሺܶ‫ ∗ ݔ‬, ‫ݔ‬௡ାଵ ሻ + ݀ሺ‫ݔ‬௡ାଵ , ‫ ∗ ݔ‬ሻ = ݀ሺܶ‫ ∗ ݔ‬, ܶ‫ݔ‬௡ ሻ + ݀ሺ‫ݔ‬௡ାଵ , ‫ ∗ ݔ‬ሻ


(2.8) (2.9)

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Letting ݊ → ∞, we get

≤ ߬ ൫݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ + ݀ሺ‫ݔ‬௡ , ܶ‫ݔ‬௡ ሻ൯ + ݀ሺ‫ݔ‬௡ାଵ , ‫ ∗ ݔ‬ሻ

= ߬ ൫݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ + ݀ሺ‫ݔ‬௡ , ‫ݔ‬௡ାଵ ሻ൯ + ݀ሺ‫ݔ‬௡ାଵ , ‫ ∗ ݔ‬ሻ

݀ሺܶ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ ≤ ߬ ൫݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ൯

(2.10)

Hence |݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ − ݀ሺܶ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ| ≤ 0. Thus, ݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ = 0 = ݀ሺܶ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ and so ܶ‫ ∗ ݔ = ∗ ݔ‬. Finally, to prove the last part of the theorem. Let us assume that ܶ‫ ∗ ݔ = ∗ ݔ‬, ܶ‫ ∗ ݕ = ∗ ݕ‬, ‫ ∗ ݔ ≠ ∗ ݕ‬, ‫ ∗ ݔ‬, ‫ܺ ∈ ∗ ݕ‬. First we show that ݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ = ݀ሺ‫ ∗ ݕ‬, ‫ ∗ ݕ‬ሻ = 0. From condition (2.1), we have ݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ ≤ ߬൫݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ൯൫ ݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ + ݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ൯ = 2߬൫݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ൯݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ

(2.11)

ିଵ ∗ ∗ Since ߬: ℝା ଴ → ሾ0, 2 ሻ be decreasing function, so the above inequality is possible if ݀ሺ‫ ݔ‬, ‫ ݔ‬ሻ = 0. ∗ ∗ሻ Similarly we can show that ݀ሺ‫ ݕ‬, ‫ = ݕ‬0. Now we consider

݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݕ‬ሻ = ݀ሺT‫ ∗ ݔ‬, ܶ‫ ∗ ݕ‬ሻ

≤ ߬൫݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݕ‬ሻ൯൫ ݀ሺ‫ ∗ ݔ‬, ܶ ‫ ∗ ݔ‬ሻ + ݀ሺ‫ ∗ ݕ‬, ܶ‫ ∗ ݕ‬ሻ൯

(2.12)

≤ ߬൫݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݕ‬ሻ൯൫ ݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ + ݀ሺ‫ ∗ ݕ‬, ‫ ∗ ݕ‬ሻ൯

This shows that ݀ሺ‫ ∗ ݔ‬, ‫ ∗ ݕ‬ሻ = 0 and ‫ ∗ ݕ = ∗ ݔ‬, which proves the uniqueness of fixed point of ܶ. Now, we furnished an example to support our main result (Theorem 2.10).

Example 2.11 Let ܺ = ሾ0,1]. Define ݀: ܺ × ܺ → ℝା ଴ by݀ሺ‫ݔ‬, ‫ݕ‬ሻ = ‫ ݔ‬+ ‫ݕ‬. Then d is dislocated metric on ܺ and the pair ሺܺ, ݀ሻ is a complete dislocated metric space. Also, we define ߬: ሾ0, ∞ሻ → ௫ ሾ0, 2ିଵ ሻ by ߬ሺ‫ݐ‬ሻ = ሺ‫ ݐ‬+ 4ሻିଵ and ܶ: ܺ → ܺ by ܶሺ‫ݔ‬ሻ = . Obviously, ߬ is a nonnegative decreasing ଻ function. Also the map T is continuous in ܺ. For all ‫ݔ‬, ‫ܺ ∈ ݕ‬, we obtain ݀൫݂ሺ‫ݔ‬ሻ, ݂ሺ‫ݕ‬ሻ൯ ≤ ߬൫݀ሺ‫ݔ‬, ‫ݕ‬ሻ൯൫݀ሺ‫ݔ‬, ݂‫ݔ‬ሻ + ݀ሺ‫ݕ‬, ݂‫ݕ‬ሻ൯

⇒

⇒

⇒

⇒

݀ ቀ଻ , ଻ ቁ ≤ ߬ሺ‫ ݔ‬+ ‫ݕ‬ሻ. ൬݀ ቀ‫ݔ‬, ଻ቁ + ݀ ቀ‫ݕ‬, ଻ቁ൰ ௫ ௬

௫ ଻

௫

+ ଻ ≤ ௫ା௬ାସ . ቀ‫ ݔ‬+ ଻ + ‫ ݕ‬+ ଻ ቁ ௬

௫ା௬ ଻

ଵ

≤ ଻ ௫ା௬ାସ ଼ ሺ௫ା௬ሻ

‫ݔ‬+‫ݕ‬+4≤ 8

⇒

⇒

௫

௬

଻

1 ≤ 8 ቀ௫ା௬ାସቁ ଵ

‫ݔ‬+‫≤ ݕ‬4,

∀ ‫ݔ‬, ‫ܺ ∈ ݕ‬.

Clearly T satisfies the condition (2.1). Thus T satisfies all the hypotheses of Theorem 2.10 and ‫ = ∗ ݔ‬0 is the unique fixed point of T. In [25] Jungck introduced the concept of commuting maps. In [26] Jungck introduced the concept of compatible mappings which generalize the concept of commuting maps. Jungck in [27] further generalized the concept of compatible maps as follows.

Definition 2.12 Let ܵ and ܶ be two mappings from a metric space ሺܺ, ݀ሻ into itself. Then, S and T are said to be weakly compatible if they commute at their coincidence points; that is, ܵ‫ ݔܶ = ݔ‬for some ‫ ܺ ∈ ݔ‬implies ܵܶ‫ݔܵܶ = ݔ‬. Now, we have the following key lemma.

Lemma 2.13 Let ܺ be a non-empty set and the mappings ܵ, ܶ, ݂ ∶ ܺ → ܺ have a unique point of coincidence ‫ ݒ‬in X. If ሺܵ, ݂ሻ and ሺܶ, ݂ ሻ are weakly compatible, then ܵ, ܶ and ݂ have a unique common fixed point. Proof Since ‫ ݒ‬is point of coincidence S, T and f. Therefore, ‫ ݑܶ = ݑܵ = ݑ݂ = ݒ‬for some ‫ܺ ∈ ݑ‬. By weakly compatibility of ሺܵ, ݂ ሻ and ሺܶ, ݂ ሻ we have ܵ‫ ݒ݂ = ݑ݂ܵ = ݑ݂ܵ = ݒ‬andܶ‫= ݑ݂ܶ = ݒ‬


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݂ܶ‫ݒ݂ = ݑ‬. It implies that ܵ‫( ݓ = ݒ݂ = ݒܶ = ݒ‬say). Then ‫ ݓ‬is a point of coincidence of ܵ, ܶ and ݂. Therefore, ‫ ݓ = ݒ‬by uniqueness. Thus ‫ ݒ‬is a unique common fixed point of ܵ, ܶ and ݂ . Our next theorem is about a common fixed point for three self-mapping satisfying contraction type condition in the context of d-metric space. Theorem 2.14 Let (X, d) be a dislocated metric space and the mappings ܵ, ܶ, ݂ ∶ ܺ → ܺ satisfy: ݀ሺܵ‫ݔ‬, ܶ‫ݕ‬ሻ ≤ ߣ݀ሺ݂‫ݔ‬, ݂‫ݕ‬ሻ

for all ‫ݔ‬, ‫ ܺ ∈ ݕ‬where 0 ≤ ߣ < 1. If ܵሺܺሻ ∪ ܶሺܺሻ ⊆ ݂ሺܺሻ and ݂ሺܺሻ is a complete subspace of X, then S, T and f have a unique point of coincidence. Moreover if ሺܵ, ݂ሻ and ሺT, ݂ ሻ are weakly compatible, then S, T and f have a unique common fixed point. (2.13)

Proof Let ‫ݔ‬଴ ∈ ܺ be arbitrary. Choose a point ‫ݔ‬ଵ in X such that ݂‫ݔ‬ଵ = ܵ‫ݔ‬଴ . This can be done since ܵሺܺሻ ⊆ ݂ሺܺሻ. Similarly, choose a point ‫ݔ‬ଶ in X such that ݂‫ݔ‬ଶ = ܶ‫ݔ‬ଵ . Continuing this process and having chosen ‫ݔ‬௡ in X, we obtain ‫ݔ‬௡ାଵ in X such that ݂‫ݔ‬ଶ௞ାଵ = ܵ‫ݔ‬ଶ௞ , ݂‫ݔ‬ଶ௞ାଶ = ܶ‫ݔ‬ଶ௞ାଵ , ݇ = 0, 1, 2, … ..

݀ሺ݂‫ݔ‬ଶ௞ାଵ , ݂‫ݔ‬ଶ௞ାଶ ሻ = ݀ሺܵ‫ݔ‬ଶ௞ , ܶ‫ݔ‬ଶ௞ାଵ ሻ ≤ ߣ݀ሺ݂‫ݔ‬ଶ௞ , ݂‫ݔ‬ଶ௞ାଵ ሻ

From (2.13), we get Similarly,

݀ሺ݂‫ݔ‬ଶ௞ାଶ , ݂‫ݔ‬ଶ௞ାଷ ሻ = ݀ሺܵ‫ݔ‬ଶ௞ାଵ , ܶ‫ݔ‬ଶ௞ାଶ ሻ ≤ ߣ݀ሺ݂‫ݔ‬ଶ௞ାଵ , f‫ݔ‬ଶ௞ାଶ ሻ

Now by induction, we obtain for each ݇ = 0, 1, 2, … .. Let

݀ሺ݂‫ݔ‬ଶ௞ାଶ , ݂‫ݔ‬ଶ௞ାଷ ሻ ≤ ߣଶ௞ାଶ ݀ሺ݂‫ݔ‬଴ , ݂‫ݔ‬ଵ ሻ ‫ݕ‬௡ = ݂‫ݔ‬௡ , ݊ = 0, 1, 2, …

݀ሺ‫ݕ‬௡ାଵ , ‫ݕ‬௡ାଶ ሻ ≤ ߣ݀ሺ‫ݕ‬௡ , ‫ݕ‬௡ାଵ ሻ ≤ ⋯ ≤ ߣ௡ାଵ ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ

(2.14)

(2.15) (2.16)

(2.17) (2.18)

Now for all n, we have

݀ሺ‫ݕ‬௠ , ‫ݕ‬௡ ሻ ≤ ݀ሺ‫ݕ‬௡ , ‫ݕ‬௡ାଵ ሻ + ݀ሺ‫ݕ‬௡ାଵ , ‫ݕ‬௡ାଶ ሻ + ⋯ + ݀ሺ‫ݕ‬௠ିଵ , ‫ݕ‬௠ ሻ

(2.19)

Now for any m > n,

≤ ሺߣ௡ + ߣ௡ାଵ + ⋯ + ߣ௠ିଵ ሻ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ ≤

(2.20)

ఒ೙ ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ ଵିఒ

Assume that ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ > 0. Letting ݊ → +∞, ሺ‫ݕ‬௡ ሻ is a Cauchy sequence. Also, if ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ = 0, then ݀ሺ‫ݕ‬௠ , ‫ݕ‬௡ ሻ = 0 for all ݉ > ݊ and hence ሺ‫ݕ‬௡ ሻ is a Cauchy sequence in X. Since ݂ሺܺሻ is complete, there exists ‫ݑ‬, ‫ ܺ ∈ ݒ‬such that ‫ݕ‬௡ → ‫ݑ݂ = ݒ‬. Now, we show that ‫ ݒ‬is a common point of coincidence of S, T and f that is ‫ݑܶ = ݑܵ = ݑ݂ = ݒ‬. ݀ሺ݂‫ݑ‬, ܵ‫ݑ‬ሻ ≤ ݀ሺ݂‫ݑ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ݀ሺ ‫ݕ‬ଶ௡ାଶ , ܵ‫ݑ‬ሻ ≤ ݀ሺ‫ݒ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ݀ሺܶ‫ݔ‬ଶ௡ାଵ , ܵ‫ݑ‬ሻ

(2.21)

≤ ݀ሺ‫ݒ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ߣ ݀ሺ݂‫ݔ‬ଶ௡ାଵ , ݂‫ݑ‬ሻ

≤ ݀ሺ‫ݒ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ߣ ݀ሺ ‫ݕ‬ଶ௡ାଵ , ‫ݒ‬ሻ → 0 as n→ ∞

Therefore, ݀ሺ݂‫ݑ‬, ܵ‫ݑ‬ሻ = 0. Similarly,

݀ሺܵ‫ݑ‬, ݂‫ݑ‬ሻ ≤ ݀ሺܵ‫ݑ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ݀ሺ ‫ݕ‬ଶ௡ାଶ , ݂‫ݑ‬ሻ ≤ ݀ሺܵ‫ݑ‬, ܶ‫ݔ‬ଶ௡ାଵ ሻ + ݀ሺ‫ݕ‬ଶ௡ାଶ , ‫ݒ‬ሻ

(2.22)

≤ ߣ݀ሺ݂‫ݑ‬, ݂‫ݔ‬ଶ௡ାଵ ሻ + ݀ሺ‫ݕ‬ଶ௡ାଶ , ‫ݒ‬ሻ

≤ ߣ݀ሺ‫ݒ‬, ‫ݕ‬ଶ௡ାଵ ሻ + ݀ሺ ‫ݕ‬ଶ௡ାଶ , ‫ݒ‬ሻ → 0 as n→ ∞.

Hence ݂‫ݑܵ = ݑ‬. Similarly, by using


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݀ሺ݂‫ݑ‬, ܶ‫ݑ‬ሻ ≤ ݀ሺ݂‫ݑ‬, ‫ݕ‬ଶ௡ାଵ ሻ + ݀ሺ ‫ݕ‬ଶ௡ାଵ , ܶ‫ݑ‬ሻ,

we can show that ݂‫ݑܶ = ݑ‬, it implies that ‫ ݒ‬is a common point of coincidence of ܵ, ܶ and ݂. Now we show that f, S and T have unique point of coincidence. For this, assume that there exists another point ‫ ∗ ݒ‬in ܺ such that ‫ ∗ݑܶ = ∗ݑܵ = ∗ݑ݂ = ∗ ݒ‬for some ‫ ∗ݑ‬in ܺ. Now, and

݀ሺ‫ݒ‬, ‫ ∗ ݒ‬ሻ = ݀ሺܵ‫ݑ‬, ܶ‫ ∗ݑ‬ሻ ≤ ߣ݀ሺ݂‫ݑ‬, ݂‫ ∗ݑ‬ሻ ≤ ߣ݀ሺ‫ݒ‬, ‫ ∗ ݒ‬ሻ

݀ሺ‫ ∗ ݒ‬, ‫ݒ‬ሻ = ݀ሺܵ‫ ∗ݑ‬, ܶ‫ݑ‬ሻ ≤ ߣ݀ሺ݂‫ ∗ݑ‬, ݂‫ݑ‬ሻ ≤ ߣ݀ሺ‫ ∗ ݒ‬, ‫ݒ‬ሻ.

(2.23)

(2.24)

This implies that |݀ሺ‫ݒ‬, ‫ ∗ ݒ‬ሻ − ݀ሺ‫ ∗ ݒ‬, ‫ݒ‬ሻ | ≤ ߣ|݀ሺ‫ݒ‬, ‫ ∗ ݒ‬ሻ − ݀ሺ‫ ∗ ݒ‬, vሻ | that is, ሺ1 − ߣሻ |݀ሺ‫ݒ‬, ‫ ∗ ݒ‬ሻ − ݀ሺ‫ ∗ ݒ‬, ‫ݒ‬ሻ | ≤ 0. Thus ݀ሺ‫ݒ‬, ‫ ∗ ݒ‬ሻ = ݀ሺ‫ ∗ ݒ‬, ‫ݒ‬ሻ = 0 and so ‫ݒ = ∗ ݒ‬. This implies that ‫ = ∗ ݒ‬v. If ሺܵ, ݂ሻ and ሺܶ, ݂ሻ are weakly compatible, by Lemma 2.13, S, T and f have a unique common fixed point. (2.25)

Put ܵ = ܶ and ݂ = ‫ ܫ‬an identity mapping in above theorem 2.14 yields Theorem 1.22. Taking ܵ = ܶ in theorem 2.14 yields Corollary 2.15 Corollary 2.15 Let (X, d) be a complete dislocated metric space and the mapping ܶ ∶ ܺ → ܺ satisfy: ݀ሺܶ‫ݔ‬, ܶ‫ݕ‬ሻ ≤ ߣ݀ሺ݂‫ݔ‬, ݂‫ݕ‬ሻ

for all ‫ݔ‬, ‫ ܺ ∈ ݕ‬where 0 ≤ ߣ < 1. If ܶሺܺሻ ⊆ ݂ሺܺሻ and f(X) is a complete subspace of ܺ, then ܶ and ݂ have a unique point of coincidence. Moreover, if ሺܶ, ݂ ሻ is weakly compatible, then ܶ and ݂ have a unique common fixed point. (2.26)

Theorem 2.16 Let ሺܺ, ݀ሻ be a dislocated metric space and the mappings ܵ, ܶ, ݂ ∶ ܺ → ܺ satisfy: ݀ሺܵ‫ݔ‬, ܶ‫ݕ‬ሻ ≤ ߣሺ݀ሺ݂‫ݔ‬, ܵ‫ݔ‬ሻ + ݀ሺ݂‫ݕ‬, ܶ‫ݕ‬ሻሻ

for all ‫ݔ‬, ‫ ܺ ∈ ݕ‬where 0 ≤ ߣ < 1. If ܵሺܺሻ ∪ ܶሺܺሻ ⊆ ݂ሺܺሻ and f(X) is a complete subspace of ܺ, then ܵ, ܶ and ݂ have a unique point of coincidence. Moreover if ሺܵ, ݂ ሻ and ሺܶ, ݂ ሻ are weakly compatible, then S, T and f have a unique common fixed point. (2.27)

Proof Let ‫ݔ‬଴ ∈ ܺ be arbitrary. Choose a point ‫ݔ‬ଵ in X such that ݂‫ݔ‬ଵ = ܵ‫ݔ‬଴ . This can be done since ܵሺܺሻ ⊆ ݂ሺܺሻ. Similarly, choose a point xଶ in X such that ݂‫ݔ‬ଶ = ܶ‫ݔ‬ଵ . Continuing this process and having chosen ‫ݔ‬௡ in X. We obtain ‫ݔ‬௡ାଵ in X such that ݂‫ݔ‬ଶ௞ାଵ = ܵ‫ݔ‬ଶ௞ , ݂‫ݔ‬ଶ௞ାଶ = ܶ‫ݔ‬ଶ௞ାଵ , ݇ = 0, 1, 2, … ..

݀ሺ݂‫ݔ‬ଶ௞ାଵ , ݂‫ݔ‬ଶ௞ାଶ ሻ = ݀ሺܵ‫ݔ‬ଶ௞ , ܶ‫ݔ‬ଶ௞ାଵ ሻ

(2.28)

From (2.27) we get

≤ ߣሺ݀ሺ݂‫ݔ‬ଶ௞ , ܵ‫ݔ‬ଶ௞ ሻ + ݀ሺ݂‫ݔ‬ଶ௞ାଵ , ܶ‫ݔ‬ଶ௞ାଵ ሻሻ

That is, Similarly,

= ߣሺ݀ሺ݂‫ݔ‬ଶ௞ , ݂‫ݔ‬ଶ௞ାଵ ሻ + ݀ሺ݂‫ݔ‬ଶ௞ାଵ , ݂‫ݔ‬ଶ௞ାଶ ሻሻ

݀ሺ݂‫ݔ‬ଶ௞ାଵ , ݂‫ݔ‬ଶ௞ାଶ ሻ ≤

ఒ ݀ሺ݂‫ݔ‬ଶ௞ , ݂‫ݔ‬ଶ௞ାଵ ሻ ଵିఒ

݀ሺ݂‫ݔ‬ଶ௞ାଶ , ݂‫ݔ‬ଶ௞ାଷ ሻ = ݀ሺܵ‫ݔ‬ଶ௞ାଵ , ܶ‫ݔ‬ଶ௞ାଶ ሻ

≤ ߣሺ݀ሺ݂‫ݔ‬ଶ௞ାଵ , ܵ‫ݔ‬ଶ௞ାଵ ሻ + ݀ሺ݂‫ݔ‬ଶ௞ାଶ , ܶ‫ݔ‬ଶ௞ାଶ ሻሻ

That is,

(2.29)

(2.30)

= ߣሺ݀ሺ݂‫ݔ‬ଶ௞ାଵ , ݂‫ݔ‬ଶ௞ାଶ ሻ + ݀ሺ݂‫ݔ‬ଶ௞ାଶ , ݂‫ݔ‬ଶ௞ାଷ ሻሻ

݀ሺ݂‫ݔ‬ଶ௞ାଶ , ݂‫ݔ‬ଶ௞ାଷ ሻ ≤

ఒ ݀ሺ݂‫ݔ‬ଶ௞ାଵ , ݂‫ݔ‬ଶ௞ାଶ ሻ. ଵିఒ

Now by induction, we obtain for each ݇ = 0, 1, 2, … ..

݀ሺ݂‫ݔ‬ଶ௞ାଵ , ݂‫ݔ‬ଶ௞ାଶ ሻ ≤ ଵିఒ ݀ሺ݂‫ݔ‬ଶ௞ , ݂‫ݔ‬ଶ௞ାଵ ሻ

ఒ ଶ ቁ ݀ሺ݂‫ݔ‬ଶ௞ିଵ , ݂‫ݔ‬ଶ௞ ሻ ଵିఒ

≤ቀ :

ఒ


(2.31)

Page 43

Manoj Ughade & R. D. Daheriya ఒ

and

Let

ଶ௞ାଵ

≤ ቀଵିఒቁ

ଶ௞ାଶ

݀ሺ݂‫ݔ‬ଶ௞ାଵ , ݂‫ݔ‬ଶ௞ାଶ ሻ ≤ ቀଵିఒቁ ఒ

‫ݕ‬௡ = ݂‫ݔ‬௡ , ݊ = 0, 1, 2, … and

ఒ ଵିఒ

݀ሺ݂‫ݔ‬଴ , ݂‫ݔ‬ଵ ሻ

݀ሺ݂‫ݔ‬଴ , ݂‫ݔ‬ଵ ሻ =‫ݎ‬

݀ሺ‫ݕ‬௡ାଵ , ‫ݕ‬௡ାଶ ሻ ≤ ‫݀ݎ‬ሺ‫ݕ‬௡ , ‫ݕ‬௡ାଵ ሻ ≤ ⋯ ≤ ‫ ݎ‬௡ାଵ ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ

Now for all n, we have

݀ሺ‫ݕ‬௠ , ‫ݕ‬௡ ሻ ≤ ݀ሺ‫ݕ‬௡ , ‫ݕ‬௡ାଵ ሻ + ݀ሺ‫ݕ‬௡ାଵ , ‫ݕ‬௡ାଶ ሻ + ⋯ + ݀ሺ‫ݕ‬௠ିଵ , ‫ݕ‬௠ ሻ

(2.32)

(2.33)

(2.34)

Now for any m > n,

≤ ሺ‫ ݎ‬௡ + ‫ ݎ‬௡ାଵ + ⋯ + ‫ ݎ‬௠ିଵ ሻ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ ≤

(2.35)

௥೙ ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ ଵି௥

Assume that ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ > 0. Letting ݊ → ∞, ሺ‫ݕ‬௡ ሻ is a Cauchy sequence. Also, if ݀ሺ‫ݕ‬଴ , ‫ݕ‬ଵ ሻ = 0, then ݀ሺ‫ݕ‬௠ , ‫ݕ‬௡ ሻ = 0 for all ݉ > ݊ and hence ሺ‫ݕ‬௡ ሻ is a Cauchy sequence in X. Since ݂ሺܺሻ is complete, there exists ‫ݑ‬, ‫ ܺ ∈ ݒ‬such that ‫ݕ‬௡ → ‫ݑ݂ = ݒ‬. Now, we show that v is a common point of coincidence of S, T and f that is ‫ݑܶ = ݑܵ = ݑ݂ = ݒ‬. ݀ሺ݂‫ݑ‬, ܵ‫ݑ‬ሻ ≤ ݀ሺ݂‫ݑ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ݀ሺ ‫ݕ‬ଶ௡ାଶ , ܵ‫ݑ‬ሻ ≤ ݀ሺ‫ݒ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ݀ሺܵ‫ݑ‬, ܶ‫ݔ‬ଶ୬ାଵ ሻ

(2.36)

≤ ݀ሺ‫ݒ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ߣ ሺ݀ሺ݂‫ݑ‬, ܵ‫ݑ‬ሻ + ݀ሺ݂‫ݔ‬ଶ௡ାଵ , ܶ‫ݔ‬ଶ௡ାଵ ሻሻ ≤ ݀ሺ‫ݒ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ߣ ሺ݀ሺ݂‫ݑ‬, ܵ‫ݑ‬ሻ + ݀ሺ݂‫ݔ‬ଶ௡ାଵ , ݂‫ݔ‬ଶ௡ାଶ ሻሻ ≤ ݀ሺ‫ݒ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ߣ ሺ݀ሺ‫ݒ‬, ܵ‫ݑ‬ሻ + ݀ሺ ‫ݕ‬ଶ௡ାଵ , ‫ݕ‬ଶ௡ାଶ ሻሻ

Letting n→ ∞, we have

݀ሺ‫ݒ‬, ܵ‫ݑ‬ሻ ≤ ߣ ݀ሺ‫ݒ‬, ܵ‫ݑ‬ሻ

Therefore, ݀ሺ‫ݒ‬, ܵ‫ݑ‬ሻ = 0. Similarly,

݀ሺܵ‫ݑ‬, ݂‫ݑ‬ሻ ≤ ݀ሺܵ‫ݑ‬, ‫ݕ‬ଶ௡ାଶ ሻ + ݀ሺ ‫ݕ‬ଶ௡ାଶ , ݂‫ݑ‬ሻ ≤ ݀ሺܵ‫ݑ‬, ܶ‫ݔ‬ଶ௡ାଵ ሻ + ݀ሺ‫ݕ‬ଶ௡ାଶ , ‫ݒ‬ሻ

Letting n→ ∞, we have

(2.37)

(2.38)

≤ ߣሺ݀ሺ݂‫ݑ‬, ܵ‫ݑ‬ሻ + ݀ሺ݂‫ݔ‬ଶ௡ାଵ , ܶ‫ݔ‬ଶ௡ାଵ ሻሻ + ݀ሺ‫ݕ‬ଶ௡ାଶ , ‫ݒ‬ሻ

≤ ߣሺ݀ሺ‫ݒ‬, ܵ‫ݑ‬ሻ + ݀ሺ‫ݕ‬ଶ௡ାଵ , ‫ݕ‬ଶ௡ାଶ ሻሻ + ݀ሺ‫ݕ‬ଶ௡ାଶ , ‫ݒ‬ሻ

݀ሺܵ‫ݑ‬, ‫ݒ‬ሻ ≤ ߣ ݀ሺ‫ݒ‬, ܵ‫ݑ‬ሻ

Hence, |݀ሺ‫ݒ‬, ܵ‫ݑ‬ሻ − ݀ሺܵ‫ݑ‬, ‫ݒ‬ሻ| ≤ 0 and then ݀ሺ‫ݒ‬, ܵ‫ݑ‬ሻ = ݀ሺܵ‫ݑ‬, ‫ݒ‬ሻ = 0. So ‫ = ݑ݂ = ݒ‬S‫ݑ‬. Similarly, by using ݀ሺ݂‫ݑ‬, ܶ‫ݑ‬ሻ ≤ ݀ሺ݂‫ݑ‬, ‫ݕ‬ଶ௡ାଵ ሻ + ݀ሺ ‫ݕ‬ଶ௡ାଵ , ܶ‫ݑ‬ሻ,

(2.39)

we can show that ݂‫ݑܶ = ݑ‬, it implies that ‫ ݒ‬is a common point of coincidence of ܵ, ܶ and ݂. Now we show that f, S and T have unique point of coincidence. For this, assume that there exists another point ‫ ∗ ݒ‬in ܺ such that ‫ ∗ݑܶ = ∗ݑܵ = ∗ݑ݂ = ∗ ݒ‬for some ‫ ∗ݑ‬in ܺ . First we show that ݀ሺ‫ݒ‬, ‫ݒ‬ሻ = ݀ሺ‫ ∗ ݒ‬, ‫ ∗ ݒ‬ሻ = 0. From condition (2.27), we have ݀ሺ‫ݒ‬, ‫ݒ‬ሻ = ݀ሺܵ‫ݑ‬, ܶ‫ݑ‬ሻ

≤ ߣሺ݀ሺ݂‫ݑ‬, ܵ‫ݑ‬ሻ + ݀ሺ݂‫ݑ‬, ܶ‫ݑ‬ሻሻ

≤ ߣሺ݀ሺ‫ݒ‬, ‫ݒ‬ሻ + ݀ሺ‫ݒ‬, ‫ݒ‬ሻሻ = 2ߣ݀ሺ‫ݒ‬, ‫ݒ‬ሻ


(2.40)

(2.41)

Page 44


Since 0 ≤ ߣ < ଶ , so the above inequality is possible if ݀ሺ‫ݒ‬, ‫ݒ‬ሻ = 0. Similarly we can show that ݀ሺ‫ ∗ ݒ‬, ‫ ∗ ݒ‬ሻ = 0. ଵ

݀ሺ‫ݒ‬, ‫ ∗ ݒ‬ሻ = ݀ሺܵ‫ݑ‬, ܶ‫ ∗ݑ‬ሻ

Now we consider

≤ ߣሺ݀ሺ‫ݒ‬, ‫ݒ‬ሻ + ݀ሺ‫ ݒ‬, ‫ ݒ‬ሻሻ ∗

(2.42)

∗

The above inequality is possible if ݀ሺ‫ݒ‬, ‫ ∗ ݒ‬ሻ = 0. which implies that ‫∗ ݒ = ݒ‬. If ሺܵ, ݂ሻ and ሺܶ, ݂ ሻ are weakly compatible, by Lemma 2.13, ܵ, ܶ and ݂ have a unique common fixed point.

Putting ܵ = ܶ and ݂ = ‫ ܫ‬an identity mapping in the above Theorem 2.16 yields Corollary 2.17.

Corollary 2.17 Let ሺܺ, ݀ሻ be a complete dislocated metric space and the mappings ܶ, ݂ ∶ ܺ → ܺ satisfy: ݀ሺܶ‫ݔ‬, ܶ‫ݕ‬ሻ ≤ ߣሺ݀ሺ‫ݔ‬, ܶ‫ݔ‬ሻ + ݀ሺ‫ݕ‬, ܶ‫ݕ‬ሻሻ

for all ‫ݔ‬, ‫ ܺ ∈ ݕ‬where 0 ≤ ߣ < 1. Then T has a unique fixed point.

(2.43)

Putting ܵ = ܶ in the above Theorem 2.16 yields Corollary 2.18.

Corollary 2.18 Let ሺܺ, ݀ሻ be a dislocated metric space and the mappings ܵ, ܶ, ݂ ∶ ܺ → ܺ satisfy: ݀ሺܶ‫ݔ‬, ܶ‫ݕ‬ሻ ≤ ߣሺ݀ሺ݂‫ݔ‬, ܶ‫ݔ‬ሻ + ݀ሺ݂‫ݕ‬, ܶ‫ݕ‬ሻሻ

for all ‫ݔ‬, ‫ ܺ ∈ ݕ‬where 0 ≤ ߣ < 1. If ܶሺܺሻ ⊆ ݂ሺܺሻ and f(X) is a complete subspace of ܺ, then ܶ and ݂ have a unique point of coincidence. Moreover if ሺܶ, ݂ ሻ is weakly compatible, then T and f have a unique common fixed point. (2.44)

Now, we present a fixed point theorem for mappings satisfying Geraghty-type contractive conditions in dislocated quasi-metric space. Let ࣭ denotes the class of the real functions ߚ: ℝା ଴ → ሾ0,1ሻ satisfying the condition ߚሺ‫ݐ‬௡ ሻ → 1 implies ‫ݐ‬௡ → 0. An example of a function in ࣭ may be given by ߚሺ‫ݐ‬ሻ = ݁ ି௧ ሺ‫ ݐ‬+ 1ሻିଵ for ‫ > ݐ‬0 and ߚሺ0ሻ ∈ ሾ0,1ሻ . Similarly the function ߚ: ℝା defined by ߚሺ‫ݐ‬ሻ = ݁ ି௧ for ‫ > ݐ‬0 ଴ → ሾ0,1ሻ and ߚሺ0ሻ ∈ ሾ0,1ሻ , belongs to the class ࣭.

Observe that we do not assume that ߚ is continuous in any sense. We only require that if ߚ gets here one, it does so only near zero. In an attempt to generalize the Banach contraction principle, Michael A. Geraghty proved in 1973 the following. Theorem 2.19 [17]Let ܶ: ܺ → ܺ be a contraction on a complete metric space satisfying ݀ሺܶ‫ݔ‬, ܶ‫ݕ‬ሻ ≤ ߙ൫݀ሺ‫ݔ‬, ‫ݕ‬ሻ൯݀ሺ‫ݔ‬, ‫ݕ‬ሻ,

(2.45)

where ߙ ∈ ࣭ .Then for any choice of initial point ‫ݔ‬଴ , the iteration ‫ݔ‬௡ = ܶ‫ݔ‬௡ିଵ for ݊ > 0 , converges to the unique fixed point ‫ݔ‬ஶ of T in X.

Theorem 2.20 Let ሺܺ, ݀ሻ be a complete dislocated quasi-metric space and ܶ: ܺ → ܺ be a self map. Suppose that there exists ߚ ∈ ࣭ such that We then have the following theorem.

݀ሺܶ‫ݔ‬, ܶ‫ݕ‬ሻ ≤ ߚ൫݀ሺ‫ݔ‬, ‫ݕ‬ሻ൯݀ሺ‫ݔ‬, ‫ݕ‬ሻ.

(2.46)

holds for all ‫ݔ‬, ‫ ܺ ∈ ݕ‬.Then ܶ has a unique fixed point ‫ ܺ ∈ ∗ ݔ‬and for each ‫ ܺ ∈ ݔ‬the Picard sequence ሼܶ ௡ ‫ݔ‬ሽ converges to ‫ ∗ ݔ‬when ݊ → ∞.

Proof Let ‫ݔ‬଴ be arbitrary in ܺ . Define a sequence ሼ‫ݔ‬௡ ሽ ⊂ ܺ by ‫ݔ‬௡ାଵ = ܶ‫ݔ‬௡ for ݊ ∈ ℕ. If ‫ݔ‬௡బ = ‫ݔ‬௡బ ାଵ for some ݊଴ ∈ ℕ, then ‫ݔ‬௡బ is a fixed point of T, and hence the proof is completed. Thus, throughout the proof, we assume that ‫ݔ‬௡ ≠ ‫ݔ‬௡ାଵ for all ݊ ∈ ℕ. From (2.46), we get 0 < ݀ሺ‫ݔ‬௡ାଵ ‫ݔ‬௡ାଶ ሻ = ݀ሺ݂‫ݔ‬௡ , ݂‫ݔ‬௡ାଵ ሻ ≤ ߚሺ݀ሺ‫ݔ‬௡ , ‫ݔ‬௡ାଵ ሻሻ݀ሺ‫ݔ‬௡ , ‫ݔ‬௡ାଵ ሻ.

Thus, we conclude that ݀ሺ‫ݔ‬௡ାଵ x௡ାଶ ሻ < ݀ሺ‫ݔ‬௡ , ‫ݔ‬௡ାଵ ሻ for all ݊ ∈ ℕ. So, the sequence ሼ݀ሺ‫ݔ‬௡ ‫ݔ‬௡ାଵ ሻሽ is nonnegative, non-increasing and bounded from below. Hence, there exists ‫ ≥ ݎ‬0 such that


(2.47)

Page 45


݈݅݉௡→ஶ ݀ሺ‫ݔ‬௡ ‫ݔ‬௡ାଵ ሻ = ‫ݎ‬. We claim that ‫ = ݎ‬0. Suppose, on the contrary, that ‫ > ݎ‬0. Then, due to (2.47), we have ௗሺ௫೙శభ ,௫೙శమ ሻ ௗሺ௫೙ ,௫೙శభ ሻ

≤ ߚ൫݀ሺ‫ݔ‬௡ , ‫ݔ‬௡ାଵ ሻ൯ < 1.

(2.48)

which yields that ݈݅݉௡→ஶ ߚ൫݀ሺ‫ݔ‬௡ , ‫ݔ‬௡ାଵ ሻ൯ = 1 . By property of ߚ ∈ ࣭ , we derive that ݈݅݉௡→ஶ ݀ሺ‫ݔ‬௡ ‫ݔ‬௡ାଵ ሻ = 0 . We shall show that ሺ‫ݔ‬௡ ሻ is a Cauchy sequence. Suppose, on the contrary, that ሺ‫ݔ‬௡ ሻ is not a Cauchy sequence. Thus, there exists ߳ > 0 such that, for all ݇ > 0, there exist ݉ሺ݇ሻ > ݊ሺ݇ሻ > ݇ with (the smallest number satisfying the condition below) Then we have

݀൫‫ݔ‬௠ሺ௞ሻ , ‫ݔ‬௡ሺ௞ሻ ൯ ≥ ߳ and ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻ ൯ < ߳.

߳ ≤ ݀൫‫ݔ‬௠ሺ௞ሻ , ‫ݔ‬௡ሺ௞ሻ ൯

≤ ݀൫‫ݔ‬௠ሺ௞ሻ , ‫ݔ‬௠ሺ௞ሻିଵ ൯ + ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻ ൯

(2.49)

(2.50)

< ݀൫‫ݔ‬௠ሺ௞ሻ , ‫ݔ‬௠ሺ௞ሻିଵ ൯ + ߳.

Letting ݇ → ∞ in the above inequality, we have ݈݅݉௞→ஶ ݀൫‫ݔ‬௠ሺ௞ሻ , ‫ݔ‬௡ሺ௞ሻ ൯ = ߳

(2.51)

By using (2.50) and (2.51), we obtain

݈݅݉௞→ஶ ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻିଵ ൯ = ߳

(2.52)

Thus from (2.46), we have

݀൫‫ݔ‬௠ሺ௞ሻ ‫ݔ‬௡ሺ௞ሻ ൯ = ݀൫ܶ‫ݔ‬௠ሺ௞ሻିଵ , ܶ‫ݔ‬௡ሺ௞ሻିଵ ൯

≤ ߚ ቀ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻିଵ ൯ቁ ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻିଵ ൯

(2.53)

< ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻିଵ ൯

Hence, we conclude that

ௗ൫௫೘ሺೖሻ ௫೙ሺೖሻ ൯

ௗ൫௫೘ሺೖሻషభ ,௫೙ሺೖሻషభ ൯

≤ ߚ ቀ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻିଵ ൯ቁ < 1

(2.54)

keeping (2.53) and (2.54) in mind and letting ݊ → ∞ in the above inequality, we derive that

݈݅݉௞→ஶ ߚ ቀ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻିଵ ൯ቁ = 1 and so, ݈݅݉௞→ஶ ݀൫‫ݔ‬௠ሺ௞ሻିଵ , ‫ݔ‬௡ሺ௞ሻିଵ ൯ = 0. Hence, ߳ = 0, which is a contradiction. So, we conclude that ሼ‫ݔ‬௡ ሽ is a Cauchy sequence. But X is a complete dq-metric space, it follows that there exists ‫ ܺ ∈ ∗ ݔ‬such that ሼ‫ݔ‬௡ ሽ converges to ‫ ∗ ݔ‬i.e. ݈݅݉௡→ஶ ݀ሺ‫ݔ‬௡ , ‫ ∗ ݔ‬ሻ = ݈݅݉௡→ஶ ݀ሺ‫ ∗ ݔ‬, ‫ݔ‬௡ ሻ = 0. Now we will prove that ‫ ∗ ݔ‬is a fixed point of T. We have ݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ ≤ ݀ሺ‫ ∗ ݔ‬, ‫ݔ‬௡ାଵ ሻ + ݀ሺx௡ାଵ , ܶ‫ ∗ ݔ‬ሻ

≤ ݀ሺ‫ ∗ ݔ‬, ‫ݔ‬௡ାଵ ሻ + ߚሺ݀ሺ‫ݔ‬௡ , ‫ ∗ ݔ‬ሻሻ݀ሺ‫ݔ‬௡ , ‫ ∗ ݔ‬ሻ

(2.55)

→ 0 ܽ‫∞ → ݊ ݏ‬.

This implies that ݀ሺ‫ ∗ ݔ‬, ܶ‫ ∗ ݔ‬ሻ = 0. Similarly, we can show that ݀ሺܶ‫ ∗ ݔ‬, ‫ ∗ ݔ‬ሻ = 0. Hence ܶ‫ ∗ ݔ = ∗ ݔ‬. Finally to prove that last part of the theorem; let us assume that ܶ‫ ∗ ݔ = ∗ ݔ‬, ܶ‫ ∗ ݕ = ∗ ݕ‬, ‫≠ ∗ ݔ‬ ‫ ∗ ݕ‬, ‫ ∗ ݔ‬, ‫ܺ ∈ ∗ ݕ‬. Therefore, 0 < ݀ሺ‫ ∗ ݔ‬, y ∗ ሻ = dሺTx ∗ , Ty ∗ ሻ

≤ β൫dሺx ∗ , y ∗ ሻ൯dሺx ∗ , y ∗ ሻ < ݀ሺx ∗ , y ∗ ሻ

(2.56)

This is a contradiction. Hence fixed point x ∗ of T is unique in X .

Now, we present one example to illustrate above Theorem 2.20.


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Example 2.21 Let X = ሾ0,1] with complete d୯ -metric defined by dሺx, yሻ = maxሼx, yሽ .The function β: ℝା ଴ → ሾ0,1ሻ defined by βሺtሻ = ୲ାଵ , for t > 0 and βሺ0ሻ ∈ ሾ0,1ሻ. Then β ∈ ࣭. Now we ୣష౪

define a self-map T: X → X by Tሺxሻ = ଺ . Take x = 1, y = 0 and obtain that ୶

dሺT1, T0ሻ = d ቀ଺ , 0ቁ = ଺, ଵ

ଵ

β൫dሺ1,0ሻ൯dሺ1,0ሻ = βሺ1ሻ. 1 = βሺ1ሻ. 1 =

ୣషభ ଵାଵ

On the other hand, take x, y ∈ X with x ≥ y. Then dሺTx, Tyሻ = d ቀ଺ , ଺ቁ = ଺, ୶ ୷

β൫dሺx, yሻ൯dሺx, yሻ = βሺxሻ. x =

୶

୶ୣష౮ ୶ାଵ

= ଶୣ > ଺ ଵ

ଵ

≥ ଶୣ > ଺ , ∀ x ∈ ሾ0,1]. ଵ

ଵ

Hence T satisfies condition of Theorem 2.20 and T has a unique fixed point x ∗ = 0.

3. CONCLUSION

To summarize, we have explored the properties of dislocated quasi-metric spaces and dislocated metric spaces. Also discuss the difference between metric space and generalizations of metric space. We established a fixed point theorem for a self-mapping in complete dislocated metric spaces under ିଵ contractive conditions related to a decreasing map τ ∶ ℝା ଴ → ሾ0, 2 ሻ . We obtained sufficient conditions for existence of points of coincidence and common fixed points of three self mappings satisfying a contractive type conditions in dislocated metric spaces. Also we presented a fixed point theorem for mappings satisfying Geraghty-type contractive conditions in dislocated quasi-metric space. We also present some examples in support of our results. Competing Interests No conflict of interest was declared by the authors. Author’s Contribution Both authors contributed equally and significantly to writing this paper. Both authors read and approved the final manuscript. Acknowledgements The authors are grateful to the reviewers and the editor for their useful comments and suggestions.

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