Applied Mathematics and Computation 217 (2010) 4094–4099

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Some periodic point results in generalized metric spaces

Mujahid Abbas a, Talat Nazir a, Stojan Radenovic b,⇑,1

a Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistanb Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

a r t i c l e i n f o

Keywords:Fixed pointPeriodic pointGeneralized metric space

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.10.026

⇑ Corresponding author.E-mail addresses: mujahid@lums.edu.pk (M. Abb

1 Stojan Radenovic is thankful to the Ministry of S

a b s t r a c t

We prove some fixed point and periodic point theorems for a map in generalized metricspaces. An example is provided to support our result. The results presented in this papergeneralize several well known comparable results in the literature.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries

The study of fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous researchactivity. It is an obvious fact that if T is a map which has a fixed point p, then p is also a fixed point of Tn for every naturalnumber n. However the converse is false. For example, consider, X = [0,1], and T defined by Tx = 1 � x. Then T has a uniquefixed point at 1

2, but every even iterate of T is the identity map, which has every point of [0,1] as a fixed point. On the otherhand, if X = [0,p], Tx = cosx, then every iterate of T has the same fixed point as T ([4,5,10]). If a map satisfies F(T) = F(Tn) foreach n 2 N, where F(T) denotes a set of all fixed point of T, then it is said to have property P. Jeong and Rhoades [5] showedthat maps satisfying many contractive conditions have property P. Abbas and Rhoades [2] studied the same problem in conemetric spaces (see also, [6]). Rhoades and Abbas [8] considered mappings satisfying a contractive condition of integral typefor which fixed points and periodic points coincide.

Mustafa and Sims [13] generalized the concept of a metric space. Based on the notion of generalized metric spaces, Mus-tafa et.al ([12,14–16]) obtained some fixed point theorems for mappings satisfying different contractive conditions.

The aim of this paper is to initiate the study of periodic point results for mappings satisfying certain contractive condi-tions in G-metric space. Our results generalize Theorems 3.1 to 3.3 of Abbas and Rhoades [2] and several well known com-parable results ([1,3,5,7,9,11]).

Consistent with Mustafa and Sims [13], the following definitions and results will be needed in the sequel.

Definition 1.1. Let X be a nonempty set. Suppose that a mapping G : X � X � X ? R+ satisfies:

(a) G(x,y,z) = 0 if and only if x = y = z,(b) 0 < G(x,y,z) for all x, y 2 X, with x – y,(c) G(x,x,y) 6 G(x,y,z) for all x, y, z 2 X, with z – y,(d) G(x,y,z) = G(x,z,y) = G(y,z,x) = . . . (symmetry in all three variables), and(e) G(x,y,z) 6 G(x,a,a) + G(a,y,z) for all x, y, z, a 2 X.

Then G is called a G-metric on X and (X,G) is called a G-metric space.

. All rights reserved.

as), talat@lums.edu.pk (T. Nazir), sradenovic@mas.bg.ac.rs, radens@beotel.rs (S. Radenovic).cience and Environmental Protection of Serbia.

M. Abbas et al. / Applied Mathematics and Computation 217 (2010) 4094–4099 4095

Definition 1.2. A G-metric is said to be symmetric if G(x,y,y) = G(y,x,x) for all x, y 2 X.

Proposition 1.3. Every G-metric space (X,G) will define a metric space (X,dG) by

dGðx; yÞ ¼ Gðx; y; yÞ þ Gðy; x; xÞ; 8x; y 2 X: ð1:1Þ

For a symmetric G-metric space

dGðx; yÞ ¼ 2Gðx; y; yÞ; 8x; y 2 X: ð1:2Þ

However, if G is not symmetric, then it holds the inequality

32

Gðx; y; yÞ 6 dGðx; yÞ 6 3Gðx; y; yÞ; 8x; y 2 X: ð1:3Þ

Definition 1.4. Let (X,G) be a G-metric space. We say that {xn} is:

(i) a G-Cauchy sequence if, for any e > 0, there is an n0 2 N (the set of all positive integers) such that for all n, m, l P n0,G(xn,xm,xl) < 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, G(x,xn,xm) < e.

A G-metric space X is said to be complete if every G-Cauchy sequence in X is convergent in X. It is known that {xn} convergesto x 2 (X,G) if and only if G(xm,xn,x) ? 0 as n, m ?1.

Definition 1.5. Let (X,G) and (X0,G0) be G-metric spaces and let f : (X,G) ? (X0,G0) be a function, then f is said to be G-contin-uous at a point x 2 X if and only if, if is G-sequentially continuous at x; that is, whenever {xn} G-convergent to x, {f(xn)} is G-convergent to f(x). A function f is G-continuous at X if and only if it is G- continuous at each a 2 X.

2. Periodic point results

First, we prove a periodic point theorem for a Banach operator type mappings in the frame work of generalized metricspaces.

Theorem 2.1. Let X be a complete G-metric space. Let f be a continuous self map on X satisfying

Gðfx; fx; f 2xÞ 6 hGðx; x; fxÞ; ð2:1Þ

or

Gðfx; f 2x; f 2xÞ 6 hGðx; fx; fxÞ ð2:2Þ

for all x 2 X where 0 6 h < 1. If either (i) (2.1) or (2.2) holds for all x 2 X, or (ii) (2.1) or (2.2) holds with strict inequality, h = 1 andfor all x 2 X, x – fx. Then F(f) – / and f has property P.

Proof. First, we show that F(f) – /. For u 2 X, define the sequence xn by xn = fn(u). From (2.1), we have

Gðxn; xn; xnþ1Þ 6 hGðxn�1; xn�1; xnÞ 6 h2Gðxn�2; xn�1; xn�1Þ 6 � � � 6 hnGðu;u; fuÞ

for all n. Now, for any m, n with m > n,

Gðxn; xn; xmÞ 6 Gðxn; xn; xnþ1Þ þ Gðxnþ1; xnþ1; xnþ2Þ þ � � � þ Gðxm�1; xm�1; xmÞ 6 ½hn þ hnþ1 þ � � � þ hm�Gðu;u; fuÞ

6hm

1� hGðu;u; fuÞ

and so G(xn,xn,xm) ? 0 as n, m ?1. Hence xn is a Cauchy sequence in X. By the G-completeness of X, limn?1xn exists. Call thelimit p. Now

Gðf nu; f nu; f nþ1uÞ 6 hGðf n�1u; f n�1u; f nuÞ 6 h2Gðf n�2u; f n�2u; f n�1uÞ 6 � � � 6 hnGðu;u; fuÞ:

On taking limit n ?1, we obtain

Gðp;p; fpÞ 6 0

and p = fp. Therefore F(f) – /.We shall always assume that n > 1, since the statement for n = 1 is trivial. Let u 2 F(fn). Suppose that f satisfies (i).

Then

4096 M. Abbas et al. / Applied Mathematics and Computation 217 (2010) 4094–4099

Gðu;u; fuÞ ¼ Gðf ðf n�1uÞ; f ðf n�1uÞ; f 2ðf n�1uÞÞ 6 hGðf n�1u; f n�1u; f nuÞ ¼ hGðf ðf n�2uÞ; f ðf n�2uÞ; f 2ðf n�2uÞÞ

6 h2Gðf n�2u; f n�2u; f n�1uÞ 6 � � � 6 hnGðu;u; fuÞ:

Since n is a fixed and n > 1, also 0 6 h < 1, then it follows that G(u,u, fu) 6 hG(u,u, fu), that is G(u,u, fu) = 0, u = fu and F(fn) = F(f).Now suppose that f satisfies (ii). If u = fu, then there is nothing to prove. Suppose, if possible, that u – fu, then by repetition

of the same argument of case (i), we have

Gðu;u; fuÞ < Gðu;u; fuÞ;

a contradiction. Therefore, in all cases, u = fu and F(fn) = F(f). h

Theorem 2.2. Let X be a complete G-metric space. Let f be a continuous self map on X satisfying

Gðfx; f 2x; f 3xÞ 6 hGðx; fx; f 2xÞ ð2:3Þ

for all x 2 X where 0 6 h < 1. If either (i) (2.3) holds for all x 2 X, or (ii) with strict inequality, h = 1 and for all x 2 X, x – fx. ThenF(f) – / and f has property P.

Proof. For u 2 X, define the sequence xn by xn = fn(u). Therefore, for all n,

Gðxn; xnþ1; xnþ2Þ 6 hGðxn�1; xn; xnþ1Þ 6 h2Gðxn�2; xn�1; xnÞ 6 � � � 6 hnGðu; fu; f 2uÞ:

Now, for any l, m, n with l > m > n,

Gðxn; xm; xlÞ 6 Gðxn; xnþ1; xnþ1Þ þ Gðxnþ1; xnþ1; xnþ2Þ þ � � � þ Gðxl�1; xl�1; xlÞ

6 Gðxn; xnþ1; xnþ2Þ þ Gðxn; xnþ1; xnþ2Þ þ � � � þ Gðxl�2; xl�1; xlÞ 6 ½hn þ hnþ1 þ � � � þ hl�Gðu; fu; f 2uÞ

6hl

1� hGðu; fu; f 2uÞ

and so G(xn,xm,xl) ? 0 as n, m, l ?1. Hence xn is a G-Cauchy sequence in X. By the G-completeness of X, limxn exists. Call thelimit p. If (2.3) holds for all x 2 X, then

G f nu; f nþ1u; f nþ2u� �

6 hG f n�1u; f nu; f nþ1u� �

6 h2Gðf n�2u; f n�1u; f n�2uÞ 6 � � � 6 hnGðu; fu; f 2uÞ:

On taking limit n ?1, we obtain that

Gðp; fp; f 2pÞ 6 0;

and p = fp. Therefore F(f) – /.Now assume that n > 1, since the statement for n = 1 is trivial. Let u 2 F(fn). Suppose that f satisfies (i). Then

Gðu; fu; f 2uÞ ¼ G f ðf n�1uÞ; f 2ðf n�1uÞ; f 3ðf n�1uÞ� �

6 hG f n�1u; f nu; f nþ1u� �

¼ hG f ðf n�2uÞ; f 2ðf n�2uÞ; f 3ðf n�2uÞ� �

6 h2Gðf n�2u; f n�1u; f nuÞ 6 � � � 6 hnGðu; fu; f 2uÞ:

Since n > 1 is a fixed and 0 6 h < 1, it follows that G(u, fu, f2u) 6 hG(u, fu, f2u), that is G(u, fu, f2u) = 0, u = fu and F(fn) = F(f).Now suppose that f satisfies (ii). If u = fu, then there is nothing to prove. Suppose, if possible, that u – fu, then by repetition

of the same argument of case (i), we have

Gðu; fu; f 2uÞ < Gðu; fu; f 2uÞ;

a contradiction. Therefore, in all cases, u = fu and F(fn) = F(f). h

Example 2.3. Let X = [0,1] and G(x,y,z) = max(jx � yj, jy � zj, jz � xj) be a G-metric on X. Define f : X ? X as

fx ¼ ab

x;

where 0 < a < b. Consider

Gðfx; f 2x; f 3xÞ ¼maxab

x� a2

b2 x����

����; a2

b2 x� a3

b3 x����

����; a3

b3 x� ab

x����

����� �

¼ ab

x� a3

b3 x� �

¼ ab

x� a2

b2 x� �

6ab

max x� ab

x����

����; ab

x� a2

b2 x����

����; x� a2

b2 x����

����� �

¼ hGðx; fx; f 2xÞ:

Hence (2.3) is satisfied for h ¼ ab. Obviously fnx converges to 0 and f is continuous at 0 2 X. Moreover, f(0) = 0, and F(fn) = F(f).

M. Abbas et al. / Applied Mathematics and Computation 217 (2010) 4094–4099 4097

Corollary 2.4. Let X be a complete G-metric space. Let f be a continuous self map on X satisfying

Gðfx; fy; fzÞ 6 k max Gðx; y; zÞ;Gðx; fx; fyÞGðy; fy; fzÞGðx; y; zÞ ;

Gðfx; fx; yÞGðfy; fy; zÞGðx; y; zÞ ;

Gðx; fx; zÞGðy; fy; zÞ2Gðx; y; zÞ

� �ð2:4Þ

for all distinct x, y, z 2 X, and k 2 [0,1). If (2.4) holds for all x 2 X, or (ii) (2.4) holds with strict inequality, k = 1 and for all x 2 X,x – fx. Then F(f) – / and f has property P.

Proof. Taking y = fx, z = f2x in (2.4) we obtain

Gðfx; f 2x; f 3xÞ 6 k max Gðx; fx; f 2xÞ;Gðx; fx; f2xÞGðfx; f 2x; f 3xÞ

Gðx; fx; f 2xÞ ;Gðfx; fx; fxÞGðf 2x; f 2x; f 2xÞ

Gðx; fx; f 2xÞ ;Gðx; fx; f 2xÞGðfx; f 2x; f 2xÞ

2Gðx; fx; f 2xÞ

� �

¼ k max Gðx; fx; f 2xÞ;Gðfx; f 2x; f 3xÞ; 0;Gðfx; f2x; f 2xÞ2

� �¼ kGðx; fx; f 2xÞ:

Hence the result follows from Theorem 2.2. h

Corollary 2.5. Let X be a complete G-metric space. Let f be a continuous self map on X satisfying

Gðfx; fy; fzÞ 6 hGðfx; y; fyÞGðfy; y; fzÞ

Gðfx; y; zÞ þ Gðfx; y; yÞ þ kGðx; y; zÞ ð2:5Þ

for all distinct x, y, z 2 X, with h, k 2 [0,1) and h + k < 1. If either (2.5) holds for all x 2 X, or (ii) (2.5) holds with strict inequality,h + k = 1 and for all x 2 X, x – fx. Then F(f) – / and f has property P.

Proof. For y = fx, z = f2x, form (2.5),

Gðfx; f 2x; f 3xÞ 6 hGðfx; fx; fx2ÞGðf 2x; fx; f 3xÞGðfx; fx; f 2xÞ þ Gðfx; fx; fxÞ þ kGðx; fx; f 2xÞ ¼ hGðfx; f 2x; f 3xÞ þ kGðx; fx; f 2xÞ:

Thus

Gðfx; f 2x; f 3xÞ 6 k1� h

Gðx; fx; f 2xÞ;

where 0 6 k1�h < 1 and so (2.3) is satisfied. Hence the result follows from Theorem 2.2. h

Theorem 2.6. Let f be a self maps on a complete G-metric space X satisfying

Gðfx; fy; fzÞ 6 h maxfGðx; y; zÞ;Gðx; fx; fxÞ;Gðy; fy; fyÞ;Gðz; fz; fzÞ; ½Gðfx; fx; yÞ þ Gðfy; fy; zÞþ Gðy; fz; fzÞ�=3; ½Gðx; fx; fxÞ þ Gðy; fy; fyÞ þ Gðz; fz; fzÞ�=3g ð2:6Þ

for all x, y, z 2 X, where 0 6 h < 12. Then F(f) – / and f has property P.

Proof. Suppose that f satisfies condition (2.6), then for all x, y 2 X, we have

Gðfx; fy; fyÞ 6 h max Gðx; y; yÞ;Gðx; fx; fxÞ;Gðy; fy; fyÞ;Gðy; fy; fyÞ; ½Gðfx; fx; yÞ þ Gðfy; fy; yÞ þ Gðy; fy; fyÞ�3

;

�

½Gðx; fx; fxÞ þ Gðy; fy; fyÞ þ Gðy; fy; fyÞ�3

�

¼ h max Gðx; y; yÞ;Gðx; fx; fxÞ;Gðy; fy; fyÞ;Gðfx; fx; yÞ þ 2Gðfy; fy; yÞ3

;Gðx; fx; fxÞ þ 2Gðfy; fy; yÞ

3

� �:

Let x0 2 X be an arbitrary point, and define the sequence xn by xn = fn(x0). Therefore, for all n,

Gðxn; xnþ1; xnþ1Þ 6 h max Gðxn�1; xn; xnÞ;Gðxn�1; xn; xnÞ;Gðxn; xnþ1; xnþ1Þ;Gðxn; xn; xnÞ þ 2Gðxnþ1; xnþ1; xnÞ

3;

�

� Gðxn�1; xn; xnÞ þ 2Gðxnþ1; xnþ1; xnÞ3

�

¼ h max Gðxn�1; xn; xnÞ;Gðxn; xnþ1; xnþ1Þ;Gðxn�1; xn; xnÞ þ 2Gðxnþ1; xnþ1; xnÞ

3

� �

6 h½Gðxn�1; xn; xnÞ þ Gðxn; xnþ1; xnþ1Þ�;

4098 M. Abbas et al. / Applied Mathematics and Computation 217 (2010) 4094–4099

implies that

Gðxn; xnþ1; xnþ1Þ 6 kGðxn�1; xn; xnÞ;

where k ¼ h1�h. Obviously 0 6 k < 1. Thus for all n,

Gðxn; xnþ1; xnþ1Þ 6 kGðxn�1; xn; xnÞ 6 k2Gðxn�2; xn�1; xn�1Þ 6 � � � 6 knGðx0; x1; x1Þ:

Now, for any m, n with m > n,

Gðxn; xm; xmÞ 6 Gðxn; xnþ1; xnþ1Þ þ Gðxnþ1; xnþ1; xnþ2Þ þ � � � þ Gðxl�1; xl�1; xlÞ 6 ½kn þ knþ1 þ � � � þ km�Gðx0; x1; x1Þ

6km

1� kGðx0; x1; x2Þ;

and so G(xn,xm,xm) ? 0 as n, m ?1. Hence xn is a G-Cauchy sequence. By G-completeness of X, there exists u 2 X such that xn

converges to u as n ?1. We claim that fu = u. If not, then consider

Gðxn; fu; fuÞ 6 h max Gðxn�1;u;uÞ;Gðxn�1; xn; xnÞ;Gðu; fu; fuÞ;Gðu; fu; fuÞ;Gðxn; xn;uÞ þ Gðfu; fu;uÞ þ Gðfu; fu;uÞ

3;

�

�Gðxn�1; xn; xnÞ þ Gðu; fu; fuÞ þ Gðu; fu; fuÞ3

�

¼ h max Gðxn�1;u;uÞ;Gðxn�1; xn; xnÞ;Gðu; fu; fuÞ;Gðxn; xn;uÞ þ 2Gðfu; fu;uÞ

3;Gðxn�1; xn; xnÞ þ 2Gðfu; fu;uÞ

3

� �:

On taking limit n ?1, then we obtain that

Gðu; fu; fuÞ 6 h max 0;Gðu; fu; fuÞ;23

Gðu; fu; fuÞ� �

¼ hGðu; fu; fuÞ;

a contradiction, implies that F(f) – /.Assume that for n > 1, since n = 1 is trivially true. Let u 2 F(fn), Then from (2.6),

�:

Gðu; fu; fuÞ ¼ Gðf ðf n�1uÞ; f ðf nuÞ; f ðf nuÞÞ 6 h max Gðf n�1u; f nu; f nuÞ;Gðf n�1u; f nu; f nuÞ;Gðf nu; f nþ1u; f nþ1uÞ;

½Gðf nu; f nu; f nuÞ þ 2Gðf nþ1u; f nþ1u; f nuÞ�3

;½Gðf n�1u; f nu; f nuÞ þ 2Gðf nu; f nþ1u; f nþ1uÞ�

3

�

¼ h max Gðf n�1u; u;uÞ;Gðu; fu; fuÞ;2Gðfu; fu;uÞ3

;Gðf n�1u;u;uÞ þ 2Gðu; fu; fuÞ

3

� �6 h½Gðf n�1u;u;uÞ þ Gðu; fu; fuÞ

Thus for k :¼ h1�h < 1,

Gðu; fu; fuÞ 6 kGðf n�1u; u;uÞ:

Thus for

Gðu; fu; fuÞ 6 kGðf n�1u; u;uÞ ¼ kGðf n�1u; f nu; f nuÞ 6 k2Gðf n�2u; f n�1u; f n�1uÞ 6 � � � 6 knGðu; fu; fuÞ:

Since n > 1 is a fixed, it follows that G(u, fu, fu) 6 k G(u, fu, fu), that is G(u, fu, fu) = 0, u = fu. Therefore in all cases, u = fu, andhence F(fn) = F(f). h

Acknowledgment

The authors are thankful to the anonymous referees for their critical remarks which helped to improve the presentation ofthe paper.

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