Journal of Applied MathematicsVolume 2012 (2012), Article ID 856974, 7 pagesdoi:10.1155/2012/856974

Research Article

Periodic Points and Fixed Points for the Weaker -Contractive Mappings in Complete Generalized Metric Spaces

Chi-Ming Chen and W. Y. Sun

Department of Applied Mathematics, National Hsinchu University of Education, No. 521 Nanda Road, Hsinchu City 300, Taiwan

Received 19 November 2011; Accepted 14 December 2011

Academic Editor: Song Cen

Copyright © 2012 Chi-Ming Chen and W. Y. Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce the notion of weaker -contractive mapping in complete metric spaces and prove the periodic points and fixed points for this type of contraction. Our results generalize or improve many recent fixed point theorems in the literature.

1. Introduction and Preliminaries

Let be a metric space, a subset of and a map. We say is contractive if there exists such that, for all , The well-known Banach’s fixed point theorem asserts that if , is contractive and is complete, then has a unique fixed point in . It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong [2] introduced the notion of -contraction. A mapping on a metric space is called -contraction if there exists an upper semicontinuous function such that

In 2000, Branciari [3] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involving three terms instead of two. Later, many authors worked on this interesting space (e.g., [4–9]).

Let be a generalized metric space. For and , we define Branciari [3] also claimed that is a basis for a topology on , is continuous in each of the coordinates and a generalized metric space is a Hausdorff space. We recall some definitions of a generalized metric space, as follows.

Definition 1.1 (see [3]). Let be a nonempty set and a mapping such that for all and for all distinct point each of them different from and , one has(i) if and only if ;(ii);(iii) (rectangular inequality). Then is called a generalized metric space (or shortly g.m.s).

Definition 1.2 (see [3]). Let be a g.m.s, a sequence in , and . We say that is g.m.s convergent to if and only if as . We denote by as .

Definition 1.3 (see [3]). Let be a g.m.s, a sequence in , and . We say that is g.m.s Cauchy sequence if and only if for each , there exists such that for all .

Definition 1.4 (see [3]). Let be a g.m.s. Then is called complete g.m.s if every g.m.s Cauchy sequence is g.m.s convergent in .

In this paper, we also recall the notion of Meir-Keeler function (see [10]). A function is said to be a Meir-Keeler function if for each , there exists such that for with , we have . Generalization of the above function has been a heavily investigated branch research. Particularly, in [11, 12], the authors proved the existence and uniqueness of fixed points for various Meir-Keeler-type contractive functions. We now introduce the notion of weaker Meir-Keeler function , as follows.

Definition 1.5. We call a weaker Meir-Keeler function if for each , there exists such that for with , there exists such that .

2. Main Results

In the paper, we denote by the class of functions satisfying the following conditions:

() is a weaker Meir-Keeler function;

() for , ;

() for all , is decreasing;

() if , then .

And we denote by the class of functions satisfying the following conditions:

() is continuous;

() for and .

Our main result is the following.

Theorem 2.1. Let be a Hausdorff and complete g.m.s, and let be a function satisfying for all and , . Then has a periodic point in , that is, there exists such that for some .

Proof. Given , define a sequence in by Step 1. We will prove that Using inequality (2.1), we have that for each and so Since is decreasing, it must converge to some . We claim that . On the contrary, assume that . Then by the definition of weaker Meir-Keeler function , there exists such that for with , there exists such that . Since , there exists such that , for all . Thus, we conclude that . So we get a contradiction. Therefore, , that is, Using inequality (2.1), we also have that for each and so Since is decreasing, by the same proof process, we also conclude Next, we claim that is g.m.s Cauchy. We claim that the following result holds.Step 2. Claim that for every , there exists such that if then .Suppose the above statement is false. Then there exists such that for any , there are with satisfying Further, corresponding to , we can choose in such a way that it the smallest integer with and . Therefore, . By the rectangular inequality and (2.3), we have Let . Then we get On the other hand, we have Let . Then we get Using inequality (2.1), we have Letting , using the definitions of the functions and , we have which implies that . By the definition of the function , we have . So we get a contradiction. Therefore is g.m.s Cauchy.Step 3. We claim that has a periodic point in .Suppose, on contrary, has no periodic point. Then is a sequence of distinct points, that is, for all with . By Step 2, since is complete g.m.s, there exists such that . Using inequality (2.1), we have Letting , we have that is, As is Hausdorff, we have , a contradiction with our assumption that has no periodic point. Therefore, there exists such that for some . So has a periodic point in .

Following Theorem 2.1, it is easy to get the below fixed point result.

Theorem 2.2. Let be a Hausdorff and complete g.m.s, and let be a function satisfying for all , where with for all , and . Then has a unique fixed point in .

Proof. From Theorem 2.1, we conclude that has a periodic point , that is, there exists such that for some . If , then we complete the proof, that is, is a fixed point of . If , then we will show that is a fixed point of . Suppose that it is not the case, that is, . Then Using inequality (2.1), we have Using inequality (2.1), we also have Continuing this process, we conclude that which implies a contradiction. Thus, is a fixed point of .Finally, to prove the uniqueness of the fixed point, suppose are fixed points of . Then, which implies that , that is, . So we complete the proof.

Acknowledgment

This research was supported by the National Science Council of the Republic of China.

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