(EA) property in generalized metric spaces

Applied Mathematics and Computation 219 (2013) 7663–7670

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Common fixed point of power contraction mappings satisfying (E.A) property in generalized metric spaces Mujahid Abbas a, Talat Nazir b, Stojan Radenovic´ c,⇑ a

Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan c University of Belgrade, Faculty of Mechanical Engineering, Krljice Marije 16, 11 120 Beograd, Serbia b

a r t i c l e

i n f o

Keywords: Compatible maps (E.A)-property Coincidence point Common fixed point Generalized metric space

a b s t r a c t In this paper, using the setting of generalized metric spaces, existence of unique common fixed point of two pairs of power contraction mappings satisfying (E.A)-property in generalized metric spaces is established. We also provide example to support the results presented herein. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries Mustafa and Sims [21] generalized the concept of a metric in which the real number is assigned to every triplet of an arbitrary set. Based on the notion of generalized metric spaces, Mustafa et al. [22–26] obtained some fixed point theorems for mappings satisfying different contractive conditions. Chugh et al. [15] obtained some fixed point results for maps satisfying property p in G metric spaces. Saadati et al. [29] studied fixed point of contractive mappings in partially ordered G metric spaces. Shatanawi [32] obtained fixed points of U maps in G metric spaces. Radenovic´ et al. [28] proved some tripled coincidence point results in G metric spaces. Recently, Nashine et al. [27] obtained coincidence and fixed point results in ordered G metric spaces. The study of unique common fixed points of mappings satisfying strict contractive conditions has been at the center of rigorous research activity. Abbas and Rhoades [2] initiated the study of common fixed point theorems in generalized metric spaces (see also, [3,5,6]). Further results in the direction of common fixed points in generalized metric space are obtained by [4,19]. Also, see [8–11,31]. The aim of this paper is to study common fixed point of weakly compatible power contraction maps satisfying (E.A)-property in the framework of G metric spaces. Consistent with Mustafa and Sims [22], the following definitions and results will be needed in the sequel. þ

Definition 1.1. Let X be a nonempty set. Suppose that a mapping G : X  X  X ! 211d satisfies:     

G1 : G2 : G3 : G4 : G5 :

Gðx; y; zÞ ¼ 0 if x ¼ y ¼ z; 0 < Gðx; y; zÞ for all x; y; z 2 X, with x – y; Gðx; x; yÞ 6 Gðx; y; zÞ for all x; y; z 2 X, with y – z; Gðx; y; zÞ ¼ Gðx; z; yÞ ¼ Gðy; z; xÞ ¼ . . ., (symmetry in all three variables); and Gðx; y; zÞ 6 Gðx; a; aÞ þ Gða; y; zÞ for all x; y; z; a 2 X.

Then G is called a G metric on X and ðX; GÞ is called a Gmetric space. ⇑ Corresponding author. E-mail addresses: [email protected] (M. Abbas), [email protected] (T. Nazir), [email protected] (S. Radenovic´). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.12.090

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Definition 1.2. A sequence fxn g in a G metric space X is: (i) a G Cauchy sequence if, for any e > 0, there is an n0 2 N (the set of natural numbers) such that for all n; m; l P n0 ; Gðxn ; xm ; xl Þ < e, (ii) a G convergent sequence if, for any e > 0, there is an x 2 X and an n0 2 N, such that for all n; m P n0 ; Gðx; xn ; xm Þ < e. A G metric space on X is said to be G complete if every GCauchy sequence in X is G convergent in X. It is known that fxn g G converges to x 2 X if and only if Gðxm ; xn ; xÞ ! 0 as n; m ! 1. Proposition 1.3. Let X be a G metric space. Then the following are equivalent: (1) (2) (3) (4)

fxn g is G convergent to x. Gðxn ; xm ; xÞ ! 0 as n; m ! 1. Gðxn ; xn ; xÞ ! 0 as n ! 1. Gðxn ; x; xÞ ! 0 as n ! 1.

Definition 1.4. A G metric on X is said to be symmetric if Gðx; y; yÞ ¼ Gðy; x; xÞ for all x; y 2 X. Proposition 1.5. Every G metric on X induces a metric dG on X given as follows

dG ðx; yÞ ¼ Gðx; y; yÞ þ Gðy; x; xÞ;

for all x; y 2 X:

ð1:1Þ

For a symmetric G metric

dG ðx; yÞ ¼ 2Gðx; y; yÞ;

for all x; y 2 X:

ð1:2Þ

However, if G is non-symmetric, then the following inequality holds:

3 Gðx; y; yÞ 6 dG ðx; yÞ 6 3Gðx; y; yÞ; 2

for all x; y 2 X:

ð1:3Þ

It is also obvious that

Gðx; x; yÞ 6 2Gðx; y; yÞ: We not that subset A in G-metric space is bounded if it is bounded in metric dG . Now, we give an example of a non-symmetric G metric. þ

Example 1.6. Let X ¼ f1; 2g and a mapping G : X  X  X ! 211d be defined as: (x, y, z)

G(x, y, z)

(1, 1, 1), (2, 2, 2) (1, 1, 2), (1, 2, 1), (2, 1, 1) (1, 2, 2), (2, 1, 2), (2, 2, 1)

0 0.5 1.

Note that G satisfies all the axioms of a generalized metric but Gðx; x; yÞ – Gðx; y; yÞ for distinct x; y in X. Therefore G, is a non-symmetric G metric on X. Sessa [30] introduced the notion of the weak commutativity of mappings in metric spaces. Recently Abbas et al. [6] studied R weakly commuting and compatible mappings in the frame work of G metric spaces. Definition 1.7 [6]. Let X be a G metric space. Mappings f ; g : X ! X are called (i) weakly commuting if Gðfgx; fgx; gfxÞ 6 Gðfx; fx; gxÞ, for all x 2 X (ii) R weakly commuting if there exists a positive real number R such that Gðfgx; fgx; gfxÞ 6 RGðfx; fx; gxÞ holds for each x 2 X (iii) compatible if, whenever a sequence fxn g in X is such that ffxn g and fgxn g are G convergent to some t 2 X, then lim Gðfgxn ; fgxn ; gfxn Þ ¼ 0 (iv) noncompatible if there exists at least one sequence n!1 fxn g in X such that ffxn g and fgxn g are G convergent to some t 2 X, but lim Gðfgxn ; fgxn ; gfxn Þ is either nonzero or does not n!1 exist. Self mappings f and g on X are said to be weakly compatible if fx ¼ gx implies fgx ¼ gfx ([18]). Thus f and g are weakly compatible if and only if f and g are pointwise R-weakly commuting mappings. In 2002, Aamri and Moutaawakil [1] introduced (E.A) property to obtain common fixed point of two mappings. Recently, Babu and Negash [12] employed this concept to obtain some new common fixed point results (see also [13,14,17]).

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Recently, Abbas et al. [7] studied (E.A)-property in the frame work of G metric space. Definition 1.8 [7]. Let X be a G metric space. Selfmaps f and g on X are said to satisfy the (E.A) property if there exists a sequence fxn g in X such that ffxn g and fgxn g are G convergent to some t 2 X. Example 1.9 [7]. Let X ¼ ½0; 2 be a G metric space with

Gðx; y; zÞ ¼ maxfjx  yj; jx  zj; jy  zjg: Let f ; g : X ! X be defined by

( fx ¼

2  x; 2x ; 2

 x 2 ½0; 1;   x 2 ð1; 2; 

and

( gx ¼

3x ; 2 x ; 2

 x 2 ½0; 1;   x 2 ð1; 2: 

n n Let us consider a decreasing sequence fxn g in X such that xn ! 1. Then gxn ! 12 ; fxn ! 12 ; gfxn ¼ 4þx ! 54 and fgxn ¼ 4x ! 32. So 4 2 fand g are noncompatible. Note that, there exists a sequence fxn g in X such that lim fxn ¼ lim gxn ¼ 1 2 X, take xn ¼ 1 for each n!1 n!1 n 2 N. Hence f and g satisfy (E.A) property.  Abbas and Rhoades [2] initiated the study of common fixed point for weakly compatible mapping in generalized metric spaces.

Definition 1.10 [2]. Let f and g be self maps of a set X. If w ¼ fx ¼ gx for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. Proposition 1.11 [2]. Let f and g be weakly compatible self maps of a set X. If f and g have a unique point of coincidence w ¼ fx ¼ gx, then w is the unique common fixed point of f and g. Definition 1.12 [16]. The control functions w and / are defined as (a) w : ½0; 1Þ ! ½0; 1Þ is a continuous nondecreasing function with wðtÞ ¼ 0 if and only if t ¼ 0, (b) / : ½0; 1Þ ! ½0; 1Þ is a lower semicontinuous function with /ðtÞ ¼ 0 if and only if t ¼ 0. 2. Common fixed point theorems In this section, we obtain some unique common fixed point results for two pairs of power contraction mappings such that one pair is satisfying (E.A)-property in the frame work of a generalized metric spaces. We start with the following result. Theorem 2.1. Let X be a G metric space and f ; g; S; T : X ! X be mappings with fX # TX and gX # SX such that

wðmaxfGd ðfx; gy; gyÞ; Gd ðfx; fx; gyÞgÞ 6 wðM x;y ðf ; g; S; TÞÞ  /ðM x;y ðf ; g; S; TÞÞ;

ð2:1Þ

hold for all x; y 2 X, where w and / are control functions and d 2 ð0; 1Þ and

Mx;y ðf ; g; S; TÞ ¼ maxfGd ðSx; Ty; TyÞ; Gd ðSx; Sx; TyÞ; Gd ðfx; Sx; SxÞ; Gd ðfx; fx; SxÞ; Gd ðTy; gy; gyÞ; Gd ðTy; Ty; gyÞ; Gd ðfx; Ty; TyÞ; Gd ðSx; gy; gyÞ; Gd ðfx; fx; TyÞ; Gd ðSx; Sx; gyÞg

ð2:2Þ

Suppose that one of the pairs ðf ; SÞ and ðg; T Þ satisfies (E.A)-property and one of the subspace f ðX Þ; g ðX Þ; SðX Þ; T ðX Þ is closed in X. If for every sequence fyn g in X, one of the following conditions hold: (a) fgyn g is bounded in case ðf ; SÞ satisfies (E.A)-property (b) ffyn g is bounded in case ðg; T Þ satisfies (E.A)-property. Then, the pairs ðf ; SÞ and ðg; TÞ have a common point of coincidence in X. Moreover, if the pairs ðf ; SÞ and ðg; T Þ are weakly compatible, then f ; g; S and T have a unique common fixed point. Proof. Suppose that the pair ðf ; SÞ satisfies (E.A)-property, there exists a sequence fxn g in X satisfying limn!1 fxn ¼ limn!1 Sxn ¼ q for some q 2 X. As fX # TX, there exists a sequence fyn g in X such that fxn ¼ Tyn . As fgyn g is bounded, n ! 1 lim sup Gðfxn ; gyn ; gyn Þ and n ! 1 lim sup GðSxn ; gyn ; gyn Þ are finite numbers. Note that

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jGðfxn ; gyn ; gyn Þ  GðSxn ; gyn ; gyn Þj 6 2Gðfxn ; Sxn ; Sxn Þ: Since Gðfxn ; Sxn ; Sxn Þ ! 0 as n ! 1, therefore n ! 1 lim sup Gðfxn ; gyn ; gyn Þ ¼ n ! 1 lim sup GðSxn ; gyn ; gyn Þ. Indeed, using

n ! 1 lim sup Gðfxn ; gyn ; gyn Þ ¼ n ! 1 lim sup GðSxn ; gyn ; gyn Þ ¼ l; for some l P 0, we obtain subsequences fxnk g and fynk g such that GðSxnk ; gynk ; gynk Þ and Gðfxnk ; gynk ; gynk Þ are G convergent to l. Replacing x by xnk and y by ynk in Eq. (2.2), we have

Mxnk ;yn ðf ; g; S; TÞ ¼ maxfGd ðSxnk ; fxnk ; fxnk Þ; Gd ðSxnk ; Sxnk ; fxnk Þ; Gd ðfxnk ; Sxnk ; Sxnk Þ; Gd ðfxnk ; fxnk ; Sxnk Þ; k

Gd ðTynk ; gynk ; gynk Þ; Gd ðTynk ; Tynk ; gynk Þ; Gd ðfxnk ; Tynk ; Tynk Þ; Gd ðSxnk ; gynk ; gynk Þ; Gd ðfxnk ; fxnk ; Tynk Þ; Gd ðSxnk ; Sxnk ; gynk Þg; which on taking limit as k ! 1 implies that d

d

d

d

d

lim Mðxnk ; ynk ; ynk Þ ¼ maxfGd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðq; q; qÞ; l ; l ; 0; l ; 0; l g ¼ l :

n!1

ð2:3Þ

Now from Eq. (2.1) we have

wðmaxfGd ðfxnk ; gynk ; gynk Þ; Gd ðfxnk ; fxnk ; gynk ÞgÞ; 6 wðM xnk ;yn ðf ; g; S; TÞÞ  /ðM xnk ;yn ðf ; g; S; TÞÞ; k

k

ð2:4Þ

which on taking upper limit gives d

d

d

wðl Þ 6 wðl Þ  /ðl Þ; d

and so l ¼ 0 for d 2 ð0; 1Þ. Thus, limn!1 Gðfxn ; gyn ; gyn Þ ¼ limn!1 GðSxn ; gyn ; gyn Þ ¼ 0 and so limn!1 yn ¼ q. If TðXÞ is a closed subspace of X. Then, there exists a p in X such that q ¼ Tp. From Eq. (2.1), we have

wðmaxfGd ðfxn ; gp; gpÞ; Gd ðfxn ; fxn ; gpÞgÞ 6 wðM xn ;p ðf ; g; S; TÞÞ  /ðM xn ;p ðf ; g; S; TÞÞ;

ð2:5Þ

where

Mxn ;p ðf ; g; S; TÞ ¼ maxfGd ðSxn ; Tp; TpÞ; Gd ðSxn ; Sxn ; TpÞ; Gd ðfxn ; Sxn ; Sxn Þ; Gd ðfxn ; fxn ; Sxn Þ; Gd ðTp; gp; gpÞ; Gd ðTp; Tp; gpÞ; Gd ðfxn ; Tp; TpÞ; Gd ðSxn ; gp; gpÞ; Gd ðfxn ; fxn ; TpÞ; Gd ðSxn ; Sxn ; gpÞg ¼ maxfGd ðSxn ; q; qÞ; Gd ðSxn ; Sxn ; qÞ; Gd ðfxn ; Sxn ; Sxn Þ; Gd ðfxn ; fxn ; Sxn Þ; Gd ðq; gp; gpÞ; Gd ðq; q; gpÞ; Gd ðfxn ; q; qÞ; Gd ðSxn ; gp; gpÞ; Gd ðfxn ; fxn ; qÞ; Gd ðSxn ; Sxn ; gpÞg; and so lim Mxn ;p ðf ; g; S; TÞ ¼ maxfGd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðq; gp; gpÞ; Gd ðq; q; gpÞ; Gd ðq; q; qÞ; Gd ðq; gp; gpÞ; Gd ðq; q; qÞ; Gd ðq; q; gpÞg

n!1

¼ maxfGd ðq; gp; gpÞ; Gd ðq; gp; gpÞg:

ð2:6Þ

Hence, we have

wðmaxfGd ðq; gp; gpÞ; Gd ðq; q; gpÞgÞ 6 wðmaxfGd ðq; gp; gpÞ; Gd ðq; q; gpÞgÞ  /ðmaxfGd ðq; gp; gpÞ; Gd ðq; q; gpÞgÞ; and /ðmaxfGd ðq; gp; gpÞ; Gd ðq; q; gpÞgÞ 6 0 . Hence maxfGd ðq; gp; gpÞ; Gd ðq; q; gpÞg ¼ 0 implies that gp ¼ q; that is, p is the coincidence point of pair ðg; TÞ. As g ðX Þ # SðX Þ, there exists a point u in X such that q ¼ Su. We claim that Su ¼ fu. From Eq. (2.1), we get

wðmaxfGd ðfu; gp; gpÞ; Gd ðfu; fu; gpÞgÞ 6 wðM u;p ðf ; g; S; TÞÞ  /ðMu;p ðf ; g; S; TÞÞ;

ð2:7Þ

where

Mu;p ðf ; g; S; TÞ ¼ maxfGd ðSu; Tp; TpÞ; Gd ðSu; Su; TpÞ; Gd ðfu; Su; SuÞ; Gd ðfu; fu; SuÞ; Gd ðTp; gp; gpÞ; Gd ðTp; Tp; gpÞ; Gd ðfu; Tp; TpÞ; Gd ðSu; gp; gpÞ; Gd ðfu; fu; TpÞ; Gd ðSu; Su; gpÞg ¼ maxfGd ðSu; Su; SuÞ; Gd ðSu; Su; SuÞ; Gd ðfu; Su; SuÞ; Gd ðfu; fu; SuÞ; Gd ðSu; Su; SuÞ; Gd ðSu; Su; SuÞ; Gd ðfu; Su; SuÞ; Gd ðSu; Su; SuÞ; Gd ðfu; fu; SuÞ; Gd ðSu; Su; SuÞg ¼ maxfGd ðfu; Su; SuÞ; Gd ðfu; Su; SuÞg:

ð2:8Þ

Hence, we have

wðmaxfGd ðfu; Su; SuÞ; Gd ðfu; fu; SuÞgÞ 6 wðmaxfGd ðfu; Su; SuÞ; Gd ðfu; Su; SuÞgÞ  /ðmaxfGd ðfu; Su; SuÞ; Gd ðfu; Su; SuÞgÞ; which implies /ðmaxfGd ðfu; Su; SuÞ; Gd ðfu; Su; SuÞgÞ 6 0. Hence

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maxfGd ðfu; Su; SuÞ; Gd ðfu; Su; SuÞg ¼ 0 and fu ¼ Su, so that u is the coincidence point of pair ðf ; SÞ. Thus fu ¼ Su ¼ Tp ¼ gp ¼ q. Now, weakly compatibility of pairs ðf ; SÞ and ðg; T Þ give that fq ¼ Sq and Tq ¼ gq. From Eq. (2.1), we have

wðmaxfGd ðfq; q; qÞ; Gd ðfq; fq; qÞgÞ ¼ wðmaxfGd ðfq; gp; gpÞ; Gd ðfq; fq; gpÞgÞ 6 wðM q;p ðf ; g; S; TÞÞ  /ðM q;p ðf ; g; S; TÞÞ;

ð2:9Þ

where

Mq;p ðf ; g; S; TÞ ¼ maxfGd ðSq; Tp; TpÞ; Gd ðSq; Sq; TpÞ; Gd ðfq; Sq; SqÞ; Gd ðfq; fq; SqÞ; Gd ðTp; gp; gpÞ; Gd ðTp; Tp; gpÞ; Gd ðfq; Tp; TpÞ; Gd ðSq; gp; gpÞ; Gd ðfq; fq; TpÞ; Gd ðSq; Sq; gpÞg ¼ maxfGd ðfq; q; qÞ; Gd ðfq; fq; qÞ; Gd ðfq; fq; fqÞ; Gd ðfq; fq; fqÞ; Gd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðfq; q; qÞ; Gðfq; q; qÞ; Gðfq; fq; qÞ; Gðfq; fq; qÞg ¼ maxfGd ðfq; q; qÞ; Gd ðfq; q; qÞg:

ð2:10Þ

From Eq. (2.9), we obtain

wðmaxfGd ðfq; q; qÞ; Gd ðfq; fq; qÞgÞ 6 wðmaxfGd ðfq; q; qÞ; Gd ðfq; fq; qÞgÞ  /ðmaxfGd ðfq; q; qÞ; Gd ðfq; fq; qÞgÞ; and so /ðmaxfGd ðfq; q; qÞ; Gd ðfq; fq; qÞgÞ 6 0. Therefore maxfGd ðfq; q; qÞ; Gd ðfq; fq; qÞg ¼ 0 and fq ¼ Sq ¼ q. Similarly it can be shown that gq ¼ q. Therefore gq ¼ Tq ¼ q. To prove uniqueness of q, suppose that fp ¼ gp ¼ Sp ¼ Tp ¼ p. From Eq. (2.1)

wðmaxfGd ðq; p; pÞ; Gd ðq; q; pÞgÞ ¼ wðmaxfGd ðfq; gp; gpÞ; Gd ðfq; fq; gpÞgÞ 6 wðMq;p ðf ; g; S; TÞÞ  /ðMq;p ðf ; g; S; TÞÞ;

ð2:11Þ

where

Mq;p ðf ; g; S; TÞ ¼ maxfGd ðSq; Tp; TpÞ; Gd ðSq; Sq; TpÞ; Gd ðfq; Sq; SqÞ; Gd ðfq; fq; SqÞ; Gd ðTp; gp; gpÞ; Gd ðTp; Tp; gpÞ; Gd ðfq; Tp; TpÞ; Gd ðSq; gp; gpÞ; Gd ðfq; fq; TpÞ; Gd ðSq; Sq; gpÞg ¼ maxfGd ðq; p; pÞ; Gd ðq; q; pÞ; Gd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðq; q; qÞ; Gd ðq; p; pÞ; Gd ðq; p; pÞGd ðq; q; pÞ; Gd ðq; q; pÞg ¼ maxfGd ðq; p; pÞ; Gd ðq; q; pÞg: Thus from Eq. (2.11), we obtain

wðmaxfGd ðq; p; pÞ; Gd ðq; q; pÞgÞ 6 wðmaxfGd ðq; p; pÞ; Gd ðq; q; pÞgÞ  /ðmaxfGd ðq; p; pÞ; Gd ðq; q; pÞgÞ; which implies that maxfGd ðq; p; pÞ; Gd ðq; q; pÞg ¼ 0 and so q ¼ p. h Example 2.2. Let X ¼ f0; 1; 2g be a set with G metric defined by ðx; y; zÞ

Gðx; y; zÞ

ð0; 0; 0Þ; ð1; 1; 1Þ; ð2; 2; 2Þ (0, 0, 1), (0, 1, 0), (1, 0, 0) (0, 1, 1), (1, 0, 1), (1, 1, 0) ð1; 2; 2Þ; ð2; 1; 2Þ; ð2; 2; 1Þ (0, 0, 2), (0, 2, 0), (2, 0, 0) (0, 2, 2), (2, 0, 2), (2, 2, 0) (1, 1, 2), (1, 2, 1), (2, 1, 1) (0, 1, 2), (0, 2, 1), (1, 0, 2) (1, 2, 0), (2, 0, 1), (2, 1, 0)

0 2 4 6

8

Note that G is a non-symmetric as Gð1; 2; 2Þ – Gð1; 1; 2Þ (see [20]). Let f ; g; S; T : X ! X be defined by x

f ðxÞ

gðxÞ

SðxÞ

TðxÞ

0 1 2

0 0 1

0 0 0

0 2 2

0 2 1

Clearly, f ðX Þ # T ðX Þ and g ðX Þ # SðX Þ with the pairs ðf ; SÞ and ðg; T Þ are weakly compatible. Also a pair ðg; T Þ satisfies (E.A) property, that is, there exists a sequence xn in X such that lim gxn ¼ lim Txn ¼ 0 2 X, take xn ¼ 0 for each n 2 N. Let n!1 n!1 w; / : ½0; 1Þ ! ½0; 1Þ be defined by

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(t wðtÞ ¼ 4t

and /ðtÞ ¼

3

if

;

e4t 3

;

t 2 ½0; 4Þ; if

t 2 ½4; 1Þ:

To check contractive conditions Eqs. (2.1) and (2.2) for all x; y 2 X, we consider the following cases: Note that for cases (i) x ¼ y ¼ 0, (ii) x ¼ 0; y ¼ 1, (iii) x ¼ 1; y ¼ 0, (iv) x ¼ 1; y ¼ 1, (v) x ¼ 2; y ¼ 0 and (vi) x ¼ 2; y ¼ 1, we have Gðfx; gy; gyÞ ¼ 0; Gðfx; fx; gyÞ ¼ 0 and hence Eqs. (2.1) and (2.2) are obviously satisfied for any value of d 2 ð0:08; 1Þ. Now (vii) if x ¼ 0; y ¼ 2, then fx ¼ 1; gy ¼ 0; Sx ¼ 2; Ty ¼ 0.

wðmaxfGd ðfx; gy; gyÞ; Gd ðfx; fx; gyÞgÞ ¼ 4 maxfGd ð1; 0; 0Þ; Gd ð1; 0; 0Þg ¼ 4ð2d Þ < ¼

11 d 11 d ð6 Þ ¼ G ð2; 0; 0Þ 3 3

11 d 11 G ðSx; Ty; TyÞ 6 M x;y ðf ; g; S; TÞ ¼ wðM x;y ðf ; g; S; TÞÞ  /ðM x;y ðf ; g; S; TÞÞ: 3 3

(viii) For x ¼ 1; y ¼ 2; then fx ¼ 1; gy ¼ 0; Sx ¼ 2; Ty ¼ 2.

  11 11 d wðmaxfGd ðfx; gy; gyÞ; Gd ðfx; fx; gyÞgÞ ¼ 4 maxfGd ð1; 0; 0Þ; Gd ð1; 0; 0Þg ¼ 4 2d < ð6d Þ ¼ G ð2; 0; 0Þ 3 3 11 d 11 ¼ G ðTy; gy; gyÞ 6 Mx;y ðf ; g; S; TÞ ¼ wðM x;y ðf ; g; S; TÞÞ  /ðMx;y ðf ; g; S; TÞÞ: 3 3 (ix) Now when x ¼ 2; y ¼ 2, then fx ¼ 1; gy ¼ 0; Sx ¼ 2; Ty ¼ 1.

wðmaxfGd ðfx; gy; gyÞ; Gd ðfx; fx; gyÞgÞ ¼ 4 maxfGd ð1; 0; 0Þ; Gd ð1; 0; 0Þg ¼ 4ð2d Þ < ¼

11 d 11 d ð8 Þ ¼ G ð2; 1; 1Þ 3 3

11 d 11 G ðSx; Ty; TyÞ 6 M x;y ðf ; g; S; TÞ ¼ wðM x;y ðf ; g; S; TÞÞ  /ðM x;y ðf ; g; S; TÞÞ 3 3

Thus all of the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique common fixed point of f ; g; S and T. h As two noncompatible selfmappings on Gmetric space X satisfy the (E.A) property, we obtain the following result. Corollary 2.3. Let X be a G metric space and f ; g; S; T : X ! X be mappings with fX # TX and gX # SX such that

wðmaxfGd ðfx; gy; gyÞ; Gd ðfx; fx; gyÞgÞ 6 wðM x;y ðf ; g; S; TÞÞ  /ðM x;y ðf ; g; S; TÞÞ;

ð2:12Þ

hold for all x; y 2 X, where w and / are control functions and d 2 ð0; 1Þ and

Mx;y ðf ; g; S; TÞ ¼ maxfGd ðSx; Ty; TyÞ; Gd ðSx; Sx; TyÞ; Gd ðfx; Sx; SxÞ; Gd ðfx; fx; SxÞ; Gd ðTy; gy; gyÞ; Gd ðTy; Ty; gyÞ; Gd ðfx; Ty; TyÞ; Gd ðSx; gy; gyÞ; Gd ðfx; fx; TyÞ; Gd ðSx; Sx; gyÞg

ð2:13Þ

Suppose that one of the pairs ðf ; SÞand ðg; T Þis noncompatible and one of the subspace f ðX Þ; g ðX Þ; SðX Þ; T ðX Þis closed in X. If for every sequence fyn g in X, one of the following conditions hold: (a) fgyn g is bounded in case ðf ; SÞ is noncompatible (b) ffyn g is bounded in case ðg; T Þ is noncompatible. Then, the pairs ðf ; SÞ and ðg; TÞ have a common point of coincidence in X. Moreover, if the pairs ðf ; SÞ and ðg; T Þare weakly compatible, then f ; g; S and T have a unique common fixed point. If we take wðt Þ ¼ t and /ðtÞ ¼ ð1  hÞt for h 2 ½0; 1Þ in Theorem 2.1, we have the following corollary. Corollary 2.4. Let X be a G metric space and f ; g; S; T : X ! X be mappings with fX # TX and gX # SX such that

maxfGd ðfx; gy; gyÞ; Gd ðfx; fx; gyÞg 6 h maxfGd ðSx; Ty; TyÞ; Gd ðSx; Sx; TyÞ; Gd ðfx; Sx; SxÞ; Gd ðfx; fx; SxÞ; Gd ðTy; gy; gyÞ; Gd ðTy; Ty; gyÞ; Gd ðfx; Ty; TyÞ;

ð2:14Þ

Gd ðSx; gy; gyÞ; Gd ðfx; fx; TyÞ; Gd ðSx; Sx; gyÞg

ð2:15Þ

hold for all x; y 2 X, where h 2 ½0; 1Þ and d 2 ð0; 1Þ. Suppose that one of the pairs ðf ; SÞand ðg; T Þsatisfies (E.A)-property and one of the subspace f ðX Þ; g ðX Þ; SðX Þ; T ðX Þis closed in X. If for every sequence fyn g in X, one of the following conditions hold: (a) fgyn g is bounded in case ðf ; SÞ satisfies (E.A)-property (b) ffyn g is bounded in case ðg; T Þ satisfies (E.A)-property.

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Then, the pairs ðf ; SÞ and ðg; TÞ have a common point of coincidence in X. Moreover, if the pairs ðf ; SÞ and ðg; T Þ are weakly compatible, then f ; g; S and T have a unique common fixed point. Corollary 2.5. Let X be a G metric space and f ; g; S; T : X ! X be mappings with fX # TX and gX # SX such that

maxfGd ðfx; gy; gyÞ; Gd ðfx; fx; gyÞg 6 a1 Gd ðSx; Ty; TyÞ þ a2 Gd ðSx; Sx; TyÞ þ a3 Gd ðfx; Sx; SxÞ þ a4 Gd ðfx; fx; SxÞ þ a5 Gd ðTy; gy; gyÞ þ a6 Gd ðTy; Ty; gyÞ þ a7 Gd ðfx; Ty; TyÞ þ a8 Gd ðSx; gy; gyÞ þ a9 Gd ðfx; fx; TyÞ þ a10 Gd ðSx; Sx; gyÞ P10

hold for all x; y 2 X, where ai P 0 with i¼1 ai < 1 and d 2 ð0; 1Þ. Suppose that one of the pairs ðf ; SÞand ðg; T Þsatisfies (E.A)-property and one of the subspace f ðX Þ; g ðX Þ; SðX Þ; T ðX Þ is closed in X. If for every sequence fyn g in X, one of the following conditions hold: (a) fgyn g is bounded in case ðf ; SÞsatisfies (E.A)-property (b) ffyn g is bounded in case ðg; T Þ satisfies (E.A)-property. Then, the pairs ðf ; SÞ and ðg; TÞ have a common point of coincidence in X. Moreover, if the pairs ðf ; SÞ and ðg; T Þare weakly compatible, then f ; g; S and T have a unique common fixed point. Remark 2.6. (1) As two noncompatible selfmappings on Gmetric space X satisfy the (E.A) property, so Corollary 2.4 and Corollary 2.5 remain true if a pair ðf ; SÞor ðg; T Þis noncompatible instead of satisfying (E.A) property. (2) A G metric can induces a metric dG given by dG ðx; yÞ ¼ Gðx; y; yÞ þ Gðx; x; yÞ. If G metric is not symmetric, the inequalities Eq. (2.1) or (2.2) does not reduce to any metric inequality with the metric dG . Hence our theorems do not reduce to fixed point problems in the corresponding metric space ðX; dG Þ.

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