S. K. Mohanta, Srikanta Mohanta SOME FIXED POINT RESULTS FOR MAPPINGS IN G-METRIC SPACES

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1 DEMONSTRATIO MATHEMATICA Vol. XLVII No S. K. Mohanta Srikanta Mohanta SOME FIXED POINT RESULTS FOR MAPPINGS IN G-METRIC SPACES Abstract. We prove a common fixed point theorem for a pair of self mappings satisfying a generalized contractive type condition in a complete G-metric space. We also deal with other fixed point results for a self mapping in the setting of generalized metric space. Our results generalize some recent results in the literature. 1. Introduction Metric fixed point theory is playing an increasing role in mathematics and applied sciences. Over the past two decades a considerable amount of research work for the development of fixed point theory have executed by several authors. There have been a number of generalizations of metric spaces such as Gähler r4 5s (called 2-metric spaces) and Dhage r2 3s (called D-metric spaces). Different authors proved that the results obtained by Gähler in 2-metric spaces are independent rather than generalizations of the corresponding results in metric spaces. However Mustafa and Sims in [13] have pointed out that most of the results claimed by Dhage and others in D-metric spaces are incorrect. They also introduced an appropriate concept of generalized metric space called G-metric space [9] and developed a new fixed point theory for various mappings in this new structure. Our aim in this study is to obtain some fixed point results in complete G-metric spaces. These results generalize some results of [11] and [14]. 2. Preliminaries We begin by briefly recalling some basic definitions and important results for G-metric spaces which will be needed in the sequel. Throughout this paper we denote by N the set of positive integers Mathematics Subject Classification: 54H25 47H10. Key words and phrases: G-metric space G-Cauchy sequence G-continuity fixed point. DOI: /dema c Copyright by Faculty of Mathematics and Information Science Warsaw University of Technology

2 180 S. K. Mohanta S. Mohanta Definition 2.1. (see [9]) Let X be a nonempty set and let G : X ˆX ˆX Ñ R` be a function satisfying the following axioms: pg 1 q Gpx y zq 0 if x y z pg 2 q 0 ă Gpx x yq for all x y P X with x y pg 3 q Gpx x yq Gpx y zq for all x y z P X with z y pg 4 q Gpx y zq Gpx z yq Gpy z xq (symmetry in all three variables) pg 5 q Gpx y zq Gpx a aq ` Gpa y zq for all x y z a P X (rectangle inequality). Then the function G is called a generalized metric or more specifically a G-metric on X and the pair px Gq is called a G-metric space. Proposition 2.1. (see [9]) Let px Gq be a G-metric space. Then for any x y z and a P X it follows that (1) if Gpx y zq 0 then x y z (2) Gpx y zq Gpx x yq ` Gpx x zq (3) Gpx y yq 2Gpy x xq (4) Gpx y zq Gpx a zq ` Gpa y zq (5) Gpx y zq 2 3 pgpx y aq ` Gpx a zq ` Gpa y zqq (6) Gpx y zq Gpx a aq ` Gpy a aq ` Gpz a aq. Definition 2.2. (see [9]) Let px Gq be a G-metric space let px n q be a sequence of points of X we say that px n q is G-convergent to x if lim nmñ8 Gpx x n x m q 0; that is for any ɛ ą 0 there exists n 0 P N such that Gpx x n x m q ă ɛ for all n m n 0. We call x as the limit of the sequence px n q and write x n Ñ x. Definition 2.3. (see [9]) Let px Gq be a G-metric space a sequence px n q is called G-Cauchy if given ɛ ą 0 there is n 0 P N such that Gpx n x m x l q ă ɛ for all n m l n 0 ; that is if Gpx n x m x l q Ñ 0 as n m l Ñ 8. Definition 2.4. (see [9]) A G-metric space px Gq is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in px Gq is G-convergent in px Gq. Proposition 2.2. In a G-metric space px Gq the following are equivalent. (1) The sequence px n q is G-Cauchy. (2) For every ɛ ą 0 there exists n 0 P N such that Gpx n x m x m q ă ɛ for all n m n 0. Definition 2.5. (see [9]) Let px Gq and px 1 G 1 q be G-metric spaces and let f : px Gq Ñ px 1 G 1 q be a function then f is said to be G-continuous at a

3 Some fixed point results for mappings in G-metric spaces 181 point a P X if given ɛ ą 0 there exists δ ą 0 such that x y P X; Gpa x yq ă δ implies G 1 pfpaq fpxq fpyqq ă ɛ. A function f is G-continuous on X if and only if it is G-continuous at all a P X. Proposition 2.3. (see [9]) Let px Gq and px 1 G 1 q be G-metric spaces then a function f : X Ñ X 1 is G-continuous at a point x P X if and only if it is G-sequentially continuous at x; that is whenever px n q is G-convergent to x pfpx n qq is G-convergent to fpxq. Proposition 2.4. (see [9]) Let px Gq be a G-metric space then the function Gpx y zq is jointly continuous in all three of its variables. 3. Main results Theorem 3.1. Let px Gq be a complete G-metric space. Suppose the mappings S T : X Ñ X satisfy & GpSpxq T pyq T pyqq. (3.1) max % GpT pxq Spyq Spyqq - a 1 Gpx y yq & Gpx T pyq T pyqq ` Gpy Spxq Spxqq. % Gpx Spyq Spyqq ` Gpy T pxq T pxqq - & Gpx Spxq Spxqq ` Gpy T pyq T pyqq. % Gpx T pxq T pxqq ` Gpy Spyq Spyqq - for all x y P X where a 1 a 2 a 3 0 with a 1 ` 2a 2 ` 2a 3 ă 1. Then S and T have a unique common fixed point in X. Proof. Let x 0 P X be arbitrary and define a sequence px n q by # Spx n 1 q if n is odd x n T px n 1 q if n is even. For any odd positive integer n P N we have by p3.1q Gpx n x n`1 x n`1 q GpSpx n 1 q T px n q T px n qq & GpSpx n 1 q T px n q T px n qq. max % GpT px n 1 q Spx n q Spx n qq -

4 182 S. K. Mohanta S. Mohanta Thus a 1 Gpx n 1 x n x n q & Gpx n 1 T px n q T px n qq ` Gpx n Spx n 1 q Spx n 1 qq. % Gpx n 1 Spx n q Spx n qq ` Gpx n T px n 1 q T px n 1 qq - & Gpx n 1 Spx n 1 q Spx n 1 qq ` Gpx n T px n q T px n qq. % Gpx n 1 T px n 1 q T px n 1 qq ` Gpx n Spx n q Spx n qq -. Gpx n x n`1 x n`1 q a 1 Gpx n 1 x n x n q which gives that ` a 2 tgpx n 1 T px n q T px n qq`gpx n Spx n 1 q Spx n 1 qqu ` a 3 tgpx n 1 Spx n 1 q Spx n 1 qq`gpx n T px n q T px n qqu a 1 Gpx n 1 x n x n q ` a 2 tgpx n 1 x n`1 x n`1 q ` Gpx n x n x n qu ` a 3 tgpx n 1 x n x n q ` Gpx n x n`1 x n`1 qu a 1 Gpx n 1 x n x n q ` a 2 tgpx n 1 x n x n q ` Gpx n x n`1 x n`1 qu ` a 3 tgpx n 1 x n x n q ` Gpx n x n`1 x n`1 qu Gpx n x n`1 x n`1 q a 1 ` a 2 ` a 3 1 a 2 a 3 Gpx n 1 x n x n q. If n is even then by p3.1q we have Gpx n x n`1 x n`1 q GpT px n 1 q Spx n q Spx n qq # + GpSpxn 1 q T px n q T px n qq max GpT px n 1 q Spx n q Spx n qq a 1 Gpx n 1 x n x n q # + Gpxn 1 T px n q T px n qq ` Gpx n Spx n 1 q Spx n 1 qq Gpx n 1 Spx n q Spx n qq ` Gpx n T px n 1 q T px n 1 qq # + Gpxn 1 Spx n 1 q Spx n 1 qq ` Gpx n T px n q T px n qq a 1 Gpx n 1 x n x n q Gpx n 1 T px n 1 q T px n 1 qq ` Gpx n Spx n q Spx n qq ` a 2 tgpx n 1 Spx n q Spx n qq ` Gpx n T px n 1 q T px n 1 qqu ` a 3 tgpx n 1 T px n 1 q T px n 1 qq ` Gpx n Spx n q Spx n qqu.

5 Some fixed point results for mappings in G-metric spaces 183 Thus Gpx n x n`1 x n`1 q a 1 Gpx n 1 x n x n q which implies that ` a 2 tgpx n 1 x n`1 x n`1 q ` Gpx n x n x n qu ` a 3 tgpx n 1 x n x n q ` Gpx n x n`1 x n`1 qu a 1 Gpx n 1 x n x n q ` a 2 tgpx n 1 x n x n q ` Gpx n x n`1 x n`1 qu ` a 3 tgpx n 1 x n x n q ` Gpx n x n`1 x n`1 qu Gpx n x n`1 x n`1 q a 1 ` a 2 ` a 3 1 a 2 a 3 Gpx n 1 x n x n q. Thus for any positive integer n p3.2q Gpx n x n`1 x n`1 q a 1 ` a 2 ` a 3 1 a 2 a 3 Gpx n 1 x n x n q. Let r a 1`a 2`a 3 1 a 2 a 3 then 0 r ă 1 since a 1 a 2 a 3 0 with a 1 `2a 2 `2a 3 ă 1. Thus p3.2q becomes p3.3q Gpx n x n`1 x n`1 q r Gpx n 1 x n x n q. By repeated application of p3.3q we obtain p3.4q Gpx n x n`1 x n`1 q r n Gpx 0 x 1 x 1 q. Then by repeated use of the rectangle inequality and (3.4) we have that for all n m P N n ă m Gpx n x m x m q Gpx n x n`1 x n`1 q ` Gpx n`1 x n`2 x n`2 q ` Gpx n`2 x n`3 x n`3 q ` ` Gpx m 1 x m x m q `r n ` r n`1 ` ` r m 1 Gpx 0 x 1 x 1 q rn 1 r Gpx 0 x 1 x 1 q. Then lim Gpx n x m x m q 0 as n m Ñ 8 since lim rn 1 r Gpx 0 x 1 x 1 q 0 as n m Ñ 8. For n m l P N pg 5 q implies that Gpx n x m x l q Gpx n x m x m q ` Gpx l x m x m q taking limit as n m l Ñ 8 we get Gpx n x m x l q Ñ 0. So px n q is a G-Cauchy sequence. By completeness of px Gq there exists u P X such that px n q is G-convergent to u.

6 184 S. K. Mohanta S. Mohanta Further by rectangle inequality and p3.1q we have Gpu T puq T puqq Gpu x 2n`1 x 2n`1 q ` Gpx 2n`1 T puq T puqq Thus we have Gpu x 2n`1 x 2n`1 q ` GpSpx 2n q T puq T puqq & GpSpx 2n q T puq T puqq. Gpu x 2n`1 x 2n`1 q ` max % GpT px 2n q Spuq Spuqq - Gpu x 2n`1 x 2n`1 q ` a 1 Gpx 2n u uq & Gpx 2n T puq T puqq ` Gpu Spx 2n q Spx 2n qq. % Gpx 2n Spuq Spuqq ` Gpu T px 2n q T px 2n qq- & Gpx 2n Spx 2n q Spx 2n qq ` Gpu T puq T puqq. % Gpx 2n T px 2n q T px 2n qq ` Gpu Spuq Spuqq- Gpu x 2n`1 x 2n`1 q ` a 1 Gpx 2n u uq ` a 2 tgpx 2n T puq T puqq ` Gpu Spx 2n q Spx 2n qqu ` a 3 tgpx 2n Spx 2n q Spx 2n qq ` Gpu T puq T puqqu. Gpu T puq T puqq Gpu x 2n`1 x 2n`1 q ` a 1 Gpx 2n u uq ` a 2 tgpx 2n T puq T puqq ` Gpu x 2n`1 x 2n`1 qu ` a 3 tgpx 2n x 2n`1 x 2n`1 q ` Gpu T puq T puqqu taking the limit as n Ñ 8 and using the fact that the function G is continuous on its variables we have Since 0 pa 2 ` a 3 q ă 1 Gpu T puq T puqq pa 2 ` a 3 q Gpu T puq T puqq. Gpu T puq T puqq 0 which implies that u T puq. Similarly we can show that Spuq u. Thus u is a common fixed point of S and T. To prove uniqueness suppose that there exists another point v in X such that v Spvq T pvq. Then Gpu v vq GpSpuq T pvq T pvqq & GpSpuq T pvq T pvqq. max % GpT puq Spvq Spvqq -

7 which gives that Some fixed point results for mappings in G-metric spaces 185 a 1 Gpu v vq & Gpu T pvq T pvqq ` Gpv Spuq Spuqq. % Gpu Spvq Spvqq ` Gpv T puq T puqq - & Gpu Spuq Spuqq ` Gpv T pvq T pvqq. % Gpu T puq T puqq ` Gpv Spvq Spvqq - a 1 Gpu v vq ` a 2 tgpu v vq ` Gpv u uqu ` a 3 tgpu u uq ` Gpv v vqu Gpu v vq a 2 1 a 1 a 2 Gpv u uq. Again by the same argument we will find that Hence Gpv u uq ˆ a 2 Gpu v vq 1 a 1 a 2 a 2 1 a 1 a 2 Gpu v vq. 2 Gpu v vq which implies that u v since 0 a 2 1 a 1 a 2 ă 1. Theorem 3.2. Let px Gq be a complete G-metric space. Suppose the mappings S T : X Ñ X satisfy & GpSpxq T pyq T pyqq. max % GpT pxq Spyq Spyqq - a 1 Gpx y yq & Gpx x T pyqq ` Gpy y Spxqq. % Gpx x Spyqq ` Gpy y T pxqq - & Gpx x Spxqq ` Gpy y T pyqq. % Gpx x T pxqq ` Gpy y Spyqq - for all x y P X where a 1 a 2 a 3 0 with a 1 ` 2a 2 ` 2a 3 ă 1. Then S and T have a unique common fixed point in X.

8 186 S. K. Mohanta S. Mohanta Proof. Let x 0 P X be arbitrary and define a sequence px n q by # Spx n 1 q if n is odd x n T px n 1 q if n is even. Then by the argument similar to that used in Theorem 3.1 we have for any positive integer n p3.5q Gpx n x n x n`1 q r n Gpx 0 x 0 x 1 q. Then by repeated use of the rectangle inequality and p3.5q we have that for all n m P N n ă m Gpx m x n x n q Gpx m x m 1 x m 1 q ` Gpx m 1 x m 2 x m 2 q ` Gpx m 2 x m 3 x m 3 q ` ` Gpx n`1 x n x n q `r n ` r n`1 ` ` r m 1 Gpx 0 x 0 x 1 q rn 1 r Gpx 0 x 0 x 1 q. So px n q becomes a G-Cauchy sequence. By completeness of px Gq there exists u P X such that px n q is G-convergent to u. As in the proof of Theorem 3.1 we can show that Gpu u T puqq pa 2 ` a 3 q Gpu u T puqq. Thus the desired conclusion follows from the same argument used in Theorem 3.1. Combining Theorems 3.1 and 3.2 we state the following theorem: Theorem 3.3. Let px Gq be a complete G-metric space. Suppose the mappings S T : X Ñ X satisfying one of the following conditions: & GpSpxq T pyq T pyqq. (3.6) max % GpT pxq Spyq Spyqq - a 1 Gpx y yq & Gpx T pyq T pyqq ` Gpy Spxq Spxqq. % Gpx Spyq Spyqq ` Gpy T pxq T pxqq - & Gpx Spxq Spxqq ` Gpy T pyq T pyqq. % Gpx T pxq T pxqq ` Gpy Spyq Spyqq -

9 Some fixed point results for mappings in G-metric spaces 187 or & GpSpxq T pyq T pyqq. (3.7) max % GpT pxq Spyq Spyqq - # + Gpx x T pyqq ` Gpy y Spxqq a 1 Gpx y yq Gpx x Spyqq ` Gpy y T pxqq & Gpx x Spxqq ` Gpy y T pyqq. % Gpx x T pxqq ` Gpy y Spyqq - for all x y P X where a 1 a 2 a 3 0 with a 1 ` 2a 2 ` 2a 3 ă 1. Then S and T have a unique common fixed point in X. As an application of Theorem 3.3 we have the following corollary. Corollary 3.1. Let px Gq be a complete G-metric space and let T : X Ñ X be a mapping satisfying one of the following conditions: GpT pxq T pyq T pyqq a 1 Gpx y yq`a 2 tgpx T pyq T pyqq`gpy T pxq T pxqqu or ` a 3 tgpx T pxq T pxqq`gpy T pyq T pyqqu GpT pxq T pyq T pyqq a 1 Gpx y yq ` a 2 tgpx x T pyqq ` Gpy y T pxqqu ` a 3 tgpx x T pxqq ` Gpy y T pyqqu for all x y P X where a 1 a 2 a 3 0 with a 1 ` 2a 2 ` 2a 3 ă 1. Then T has a unique fixed point in X. Proof. Put S T in Theorem 3.3. Remark 3.1. Putting a 1 a 3 0 in Corollary 3.1 we obtain Theorem 2.9 from [11]. Theorem 3.4. Let px Gq be a complete G-metric space and let T : X Ñ X be such that for each positive integer n p3.8q GpT n pxq T n pyq T n pyqq a n Gpx y yq for all x y P X where a n ą 0 is independent of x y. If the series ř 8 n 1 a n is convergent then T has a unique fixed point in X. Proof. Let x 0 P X be arbitrary and define a sequence px n q by x n T px n 1 q T n px 0 q for n Then by repeated use of the rectangle inequality and p3.8q we have that for all n m P N n ă m

10 188 S. K. Mohanta S. Mohanta (3.9) Gpx n x m x m q Gpx n x n`1 x n`1 q ` Gpx n`1 x n`2 x n`2 q ` Gpx n`2 x n`3 x n`3 q ` ` Gpx m 1 x m x m q m 1 ÿ r n m 1 ÿ r n m 1 ÿ r n Gpx r x r`1 x r`1 q GpT r px 0 q T r px 1 q T r px 1 qq a r Gpx 0 x 1 x 1 q. If x 1 x 0 then a fixed point is obtained. Therefore we assume that x 1 x 0. Let ř k be a positive integer such that k ą Gpx 0 x 1 x 1 q. Since the series 8 n 1 a n is convergent for ɛ ą 0 arbitrary there exists a positive integer n 0 such that m 1 ÿ r n a r ă ɛ k for m ą n n 0. Then for m ą n n 0 we have from p3.9q Gpx n x m x m q ɛ k Gpx 0 x 1 x 1 q ă ɛ. By Proposition 2.2 the sequence px n q becomes a G-Cauchy sequence. Using the completeness of px Gq there exists u P X such that px n q is G-convergent to u. But by pg 5 q and p3.8q we have Gpu T puq T puqq Gpu x n`1 x n`1 q ` Gpx n`1 T puq T puqq Gpu x n`1 x n`1 q ` GpT px n q T puq T puqq Gpu x n`1 x n`1 q ` a 1 Gpx n u uq taking the limit as n Ñ 8 and using the fact that the function G is continuous on its variables we have Gpu T puq T puqq 0 which implies that u T puq and u becomes a fixed point of T. For uniqueness suppose that v u is such that T pvq v. Then for any positive integer n we have Gpu v vq GpT n puq T n pvq T n pvqq a n Gpu v vq. Since by G 2 Gpu v vq ą 0 it must be the case that a n 1 for all n. So a n can not tend to zero and this contradiction shows that u v. As an application of Theorem 3.4 we have the following Corollary.

11 Some fixed point results for mappings in G-metric spaces 189 Corollary 3.2. (see [14]) Let px Gq be a complete G-metric space and let T : X Ñ X be a mapping satisfying the following condition for all x y P X p3.10q GpT pxq T pyq T pyqq k Gpx y yq where 0 k ă 1. Then T has a unique fixed point in X. Proof. For x y P X we obtain from p3.10q that GpT 2 pxq T 2 pyq T 2 pyqq k GpT pxq T pyq T pyqq k 2 Gpx y zq. Similarly for any positive integer n GpT n pxq T n pyq T n pyqq k n Gpx y yq for all x y P X. But the series ř 8 n 1 kn is convergent. Now Theorem 3.4 applies to obtain a unique fixed point of T. Theorem 3.5. Let px Gq be a complete G-metric space and let T : X Ñ X be G-continuous. Suppose that there exists a mapping Q : X Ñ r0 8q such that p3.11q Gpx T pxq T pxqq Qpxq QpT pxqq for all x P X. Then T has a fixed point in X. Proof. Let x 0 P X be arbitrary and define a sequence px n q by x n T px n 1 q for n Then for any positive integer r we have by using p3.11q that Therefore Gpx r x r`1 x r`1 q Gpx r T px r q T px r qq n 1 ÿ r 0 Qpx r q QpT px r qq Qpx r q Qpx r`1 q. Gpx r x r`1 x r`1 q n 1 ÿ r 0 rqpx r q Qpx r`1 qs Qpx 0 q Qpx n q Qpx 0 q. So the series ř 8 r 0 Gpx r x r`1 x r`1 q is convergent. Then for all n m P N n ă m we have by repeated use of the rectangle inequality that (3.12) Gpx n x m x m q Gpx n x n`1 x n`1 q ` Gpx n`1 x n`2 x n`2 q ` Gpx n`2 x n`3 x n`3 q ` ` Gpx m 1 x m x m q m 1 ÿ r n Gpx r x r`1 x r`1 q.

12 190 S. K. Mohanta S. Mohanta The convergence of the series ř 8 r 0 Gpx r x r`1 x r`1 q gives for arbitrary ɛ ą 0 there exists a positive integer n 0 such that m 1 ÿ r n Gpx r x r`1 x r`1 q ă ɛ for m ą n n 0. Then for m ą n n 0 we have from p3.12q that Gpx n x m x m q ă ɛ. By Proposition 2.2 the sequence px n q becomes a G-Cauchy sequence. Using the completeness of px Gq there exists u P X such that px n q is G-convergent to u. G-continuity of T implies that Thus u is a fixed point of T. T puq lim n T px n q lim n x n`1 u. Remark 3.2. A fixed point of T in the above theorem is not unique. The identity mapping I satisfies the condition p3.11q but a fixed point of I is not unique. References [1] R. Chugh T. Kadian A. Rani B. E. Rhoades Property P in G-metric spaces Fixed Point Theory and Applications vol Article ID pages [2] B. C. Dhage Generalised metric spaces and mappings with fixed point Bull. Calcutta Math. Soc. 84(4) (1992) [3] B. C. Dhage Generalised metric spaces and topological structure I An. Stiint. Univ. "Al. I. Cuza" Iasi Mat. 46(1) (2000) [4] S. Gahler 2-metrische Räume und ihre topologische Struktur Math. Nachr. 26 (1963) [5] S. Gahler Zur geometric 2-metrische räume Rev. Roumaine Math. Pures Appl. 40 (1966) [6] L. Gajić Z. Lozanov-Crvenković A fixed point result for mappings with contractive iterate at a point in G-metric spaces Filomat 25(2) (2011) [7] K. S. Ha Y. J. Cho A. White Strictly convex and strictly 2-convex 2-normed spaces Math. Japon. 33(3) (1988) [8] S. K. Mohanta Property P of Ćirić operators in G-metric spaces Internat. J. Math. Sci. Engg. Appl. 5 (2011) [9] Z. Mustafa B. Sims A new approach to generalized metric spaces J. Nonlinear Convex Anal. 7(2) (2006) [10] Z. Mustafa B. Sims Fixed point theorems for contractive mappings in complete G-metric spaces Fixed Point Theory and Applications vol Article ID pages [11] Z. Mustafa H. Obiedat F. Awawdeh Some fixed point theorem for mapping on complete G-metric spaces Fixed Point Theory and Applications vol Article ID pages 2008.

13 Some fixed point results for mappings in G-metric spaces 191 [12] Z. Mustafa W. Shatanawi M. Bataineh Existence of fixed point results in G-metric spaces Int. J. Math. Math. Sci. vol Article ID pages [13] Z. Mustafa B. Sims Some remarks concerning D-metric spaces in Proceedings of the International Conference on Fixed Point Theory and Applications pp Valencia Spain July [14] Z. Mustafa A new structure for generalized metric spaces with applications to fixed point theory Ph. D. thesis The University of Newcastle Callaghan Australia [15] Z. Mustafa H. Obiedat A fixed points theorem of Reich in G-metric spaces CUBO A Mathematical Journal 12(01) (2010) [16] Z. Mustafa F. Awawdeh W. Shatanawi Fixed point theorem for expansive mappings in G-metric spaces Int. J. Contemp. Math. Sci. 5(50) (2010) [17] S. V. R. Naidu K. P. R. Rao N. S. Rao On the concept of balls in a D-metric space Int. J. Math. Math. Sci. 1 (2005) [18] W. Shatanawi Fixed point theory for contractive mappings satisfying φ-maps in G-metric spaces Fixed Point Theory and Applications vol Article ID pages DEPARTMENT OF MATHEMATICS WEST BENGAL STATE UNIVERSITY BARASAT 24 PARGANS (NORTH) WEST BENGAL KOLKATA INDIA smwbes@yahoo.in Received May ; revised version February