Abstract
The purpose of this paper is to present the existence of the best period proximity point for cyclic weaker Meir-Keeler contractions and asymptotic cyclic weaker Meir-Keeler contractions in metric spaces.
1. Introduction and Preliminaries
Throughout this paper, by we denote the set of all nonnegative numbers, while is the set of all natural numbers. Let and be nonempty subsets of a metric space . Consider a mapping , is called a cyclic map if and . A point in is called a best proximity point of in if is satisfied, where , and is called a best periodic proximity point of in if is satisfied, for some . In 2005, Eldred et al. [1] proved the existence of a best proximity point for relatively nonexpansive mappings using the notion of proximal normal structure. In 2006, Eldred and Veeramani [2] proved the following existence theorem.
Theorem 1.1 (see Theoremββ3.10 in [2]). Let and be nonempty closed convex subsets of a uniformly convex Banach space. Suppose is a cyclic contraction, that is, and , and there exists such that Then there exists a unique best proximity point in . Further, for each , converges to the best proximity point.
In this paper, we also recall the notion of Meir-Keeler type mapping. A mapping is said to be a Meir-Keeler-type mapping (see [3]) if for each , there exists such that for with , we have .
In the recent, Eldred et al. [1] introduced the below notion of cyclic Meir-Keeler contraction.
Definition 1.2 (see [1]). Let be a metric space, and let and be nonempty subsets of . Then is called a cyclic Meir-Keeler contraction if the following are satisfied: (i) and ;(ii) for every , there exists such that
for all and .
In the recent, Di Bari et al. [4] proved the following best proximity point theorem.
Theorem 1.3 (see [4]). Let be a uniformly convex Banach space, and let and be nonempty subsets of . Suppose is closed and convex and is a cyclic Meir-Keeler contraction. Then there exists a unique best proximity point in . Further, for each , converges to best proximity point.
Later, many authors studied this subject, and many results on best proximity points are proved. (see, e.g., [5β10]). In this study, we will introduce the new concepts of cyclic weaker Meir-Keeler contractions and asymptotic cyclic weaker Meir-Keeler contractions in metric spaces, and the purpose of this paper is to present the existence of the best period proximity point for these contractions.
2. The Best Periodic Proximity Points for Cyclic Weaker Meir-Keeler Contractions
In this section, we first introduce the below notions of the weaker Meir-Keeler-type mapping, -mapping, and cyclic weaker Meir-Keeler contraction in metric spaces.
Definition 2.1. Let be a metric space, and . Then is called a weaker Meir-Keeler-type mapping in if for each , there exists such that for with , there exists such that .
The following provides an example of a weaker Meir-Keeler-type mapping that is not a Meir-Keeler-type mapping in a metric space.
Example 2.2. Let , and we define by If , where, , then is a weaker Meir-Keeler-type mapping that is not a Meir-Keeler-type mapping in .
Definition 2.3. Let be a metric space. A mapping is called a -mapping in if the mapping satisfies the following conditions:() is a weaker Meir-Keeler-type mapping in ;() for all , is nonincreasing;() for all , and .
The following provides two examples of a -mapping.
Example 2.4. Let and we define by Let be Then is a -mapping in .
Example 2.5. Let , and we define by If , where , , then is a -mapping in .
Definition 2.6. Let be a metric space, and let and be nonempty subsets of . Then is called a cyclic weaker Meir-Keeler contraction if the following conditions hold:(1) and ;(2) there is a -mapping in such that for all and , with ,
The following provides an example of a cyclic weaker Meir-Keeler contraction.
Example 2.7. Let and in the metric space , where . Define
Let be defined by
where , , . Then all conditions (1) and (2) of Definition 2.6 and therefore are a cyclic weaker Meir-Keeler contraction. Notice that .
Now, we are in this position to state the following results.
Lemma 2.8. Let be a metric space, and let , be nonempty subsets of . Suppose is a cyclic weaker Meir-Keeler contraction. Then holds.
Proof. Since is a cyclic weaker Meir-Keeler contraction, there is a -mapping in such that
for all.
Since is nonincreasing, hence we also conclude is nonincreasing, and it must converge to some . We claim that . On the contrary, assume that . By the definition of the weaker Meir-Keeler-type mapping , corresponding to use, there exists such that for with , there exists such that . Since , there exists such that , forall . Thus, we conclude that . So we get a contradiction. So , and so , that is, . Thus, we also conclude that .
Applying above Lemma 2.8, it is easy to conclude the following theorem.
Theorem 2.9. Let be a metric space, and let be nonempty subsets of . Suppose is a cyclic weaker Meir-Keeler contraction and if for some , the sequence converges to , then is a best periodic proximity point of in .
Proof. By the definition of the weaker Meir-Keeler-type mapping in , there exists such that for each . Since converges to, corresponding to above use, we have
Letting . Then . Thus is a best period proximity point of in .
3. The Best Periodic Proximity Points for Asymptotic Cyclic Weaker Meir-Keeler Contractions
In this section, we introduce the below notions of the asymptotic cyclic weaker Meir-Keeler-type sequence and asymptotic cyclic weaker Meir-Keeler contraction in a metric space .
Definition 3.1. Let be a metric space. A sequence in is called an asymptotic weaker Meir-Keeler-type sequence if satisfies the following conditions: for each , there exists such that for with , there exists such that ; for all and , is nonincreasing; for all , and , .
Example 3.2. Let and we define by Let be where , . Then is an asymptotic weaker Meir-Keeler-type sequence in a metric space .
Definition 3.3. Let be a metric space, and let and be nonempty subsets of . Then is an asymptotic cyclic weaker Meir-Keeler contraction if the following conditions hold:
(1) and ;(2) there is an asymptotic weaker Meir-Keeler-type sequence such that for all and , with ,
Now, we are in this position to state the following results.
Lemma 3.4. Let be a metric space and nonempty subsets of . Suppose is an asymptotic cyclic weaker Meir-Keeler contraction. Then holds.
Proof. Since is an asymptotic cyclic weaker Meir-Keeler contraction, there is an asymptotic weaker Meir-Keeler-type sequence such that
for all and , .
Since is nonincreasing, hence we also conclude is nonincreasing, and it must converge to some . We claim that . On the contrary, assume that . By the definition of asymptotic weaker Meir-Keeler-type sequence, corresponding to use, 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, , and so , that is, . Thus, we also conclude that .
Applying above Lemma 3.4, we are easy to conclude the following theorem.
Theorem 3.5. Let be a metric space and , nonempty subsets of . Suppose is an asymptotic cyclic weaker Meir-Keeler contraction, and if for some , the sequence converges to , then is a best periodic proximity point of in .
Proof. By the definition of the asymptotic weaker Meir-Keeler-type sequence , thus there exists such that for each . Since converges to , corresponding to above use, we have
Letting . Then . Thus is a best period proximity point of in .
Acknowledgments
The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper. This paper was supported by the National Science Council of the Republic of China.