- Research Article
- Open Access

# Best Proximity Points of Cyclic -Contractions on Reflexive Banach Spaces

- Sh Rezapour
^{1}, - M Derafshpour
^{1}and - N Shahzad
^{2}Email author

**2010**:946178

https://doi.org/10.1155/2010/946178

© The Author(s). 2010

**Received:**8 July 2009**Accepted:**12 January 2010**Published:**20 January 2010

## Abstract

We provide a positive answer to a question raised by Al-Thagafi and Shahzad (Nonlinear Analysis, 70 (2009), 3665-3671) about best proximity points of cyclic -contractions on reflexive Banach spaces.

## Keywords

- Banach Space
- Natural Number
- Differential Geometry
- Normed Space
- Nonempty Subset

## 1. Introduction

As a generalization of Banach contraction principle, Kirk et al. proved, in 2003, the following fixed point result; see [1].

Theorem 1.1.

Let and be nonempty closed subsets of a complete metric space . Suppose that is a map satisfying , and there exists such that for all and . Then, has a unique fixed point in .

Let and be nonempty closed subsets of a metric space and a strictly increasing map. We say that is a cyclic -contraction map [2] whenever , and

for all and , where . Also, is called a best proximity point if . As a special case, when , in which is a constant, is called cyclic contraction.

In 2005, Petru el proved some periodic point results for cyclic contraction maps [3]. Then, Eldered and Veeramani proved some results on best proximity points of cyclic contraction maps in 2006 [4]. They raised a question about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. In 2009, Al-Thagafi and Shahzad gave a positive answer to the question [2]. More precisely, they proved some results on the existence and convergence of best proximity points of cyclic contraction maps defined on reflexive (and strictly convex) Banach spaces [2, Theorems 9, 10, 11, and 12]. They also introduced cyclic -contraction maps and raised the following question in [2].

Question 1.

It is interesting to ask whether Theorems 9 and 10 (resp., Theorems 11 and 12) hold for cyclic -contraction maps where the space is only reflexive (resp., reflexive and strictly convex) Banach space.

In this paper, we provide a positive answer to the above question. For obtaining the answer, we use some results of [2].

## 2. Main Results

First, we give the following extension of [4, Proposition 3.3] for cyclic -contraction maps, where is unbounded.

Theorem 2.1.

Let be a strictly increasing unbounded map. Also, let and be nonempty subsets of a metric space , a cyclic -contraction map, and for all . Then, the sequences and are bounded.

Proof.

. Hence, . This contradiction completes the proof.

Since the proof of last result was classic, we presented it separately. Here, we provide our key result via a special proof which is a general case of Theorem 2.1.

Theorem 2.2.

Let be a strictly increasing map. Also, let and be nonempty subsets of a metric space , a cyclic -contraction map, , and for all . Then, the sequences and are bounded.

Proof.

is a constant. But, for all . This contradiction completes the proof.

Now by using this key result, we provide our main results which give positive answer to the question. Their proofs are basically due to Al-Thagafi and Shahzad [2]. However, the crucial role is played by our key result. Weak convergence of to is denoted by .

Theorem 2.3.

Proof.

Hence, .

Definition 2.4.

Theorem 2.5.

Let be a strictly increasing map. Also, let and be nonempty subsets of a reflexive Banach space such that is weakly closed and a cyclic -contraction map. Then, there exists such that provided that one of the following conditions is satisfied

(a) is weakly continuous on .

(b) satisfies the proximal property.

Proof.

If , the result follows from [2, Theorem 1]. So, we assume that . For , define for all . By Theorem 2.2, the sequence is bounded. Since is reflexive and is weakly closed, the sequence has a subsequence such that as .

(a)Since is weakly continuous on and , we have as . So as . The rest of the proof is similar to that of Theorem 2.3.

(b)By [2, Theorem 3], we have

as . Since satisfies the proximal property, we have .

Theorem 2.6.

Let be a strictly increasing map. Also, let and be nonempty closed and convex subsets of a reflexive and strictly convex Banach space and a cyclic -contraction map. If , then there exists a unique such that and .

Proof.

which is a contradiction. Since , we obtain, from the uniqueness of , that . Hence , and .

Theorem 2.7.

Let be a strictly increasing map. Also, let and be nonempty subsets of a reflexive and strictly convex Banach space such that is closed and convex and a cyclic -contraction map. Then, there exists a unique such that and provided that one of the following conditions is satisfied

(a) is weakly continuous on .

(b) satisfies the proximal property.

Proof.

which is a contradiction. The uniqueness of follows as in the proof of [2, Theorem 8].

## Declarations

### Acknowledgment

The authors express their gratitude to the referees for their helpful suggestions concerning the final version of this paper.

## Authors’ Affiliations

## References

- Kirk WA, Srinivasan PS, Veeramani P:
**Fixed points for mappings satisfying cyclical contractive conditions.***Fixed Point Theory*2003,**4**(1):79–89.MathSciNetMATHGoogle Scholar - Al-Thagafi MA, Shahzad N:
**Convergence and existence results for best proximity points.***Nonlinear Analysis. Theory, Methods & Applications*2009,**70**(10):3665–3671. 10.1016/j.na.2008.07.022MathSciNetView ArticleMATHGoogle Scholar - Petruşel G:
**Cyclic representations and periodic points.***Universitatis Babeş-Bolyai. Studia. Mathematica*2005,**50**(3):107–112.MathSciNetMATHGoogle Scholar - Eldred AA, Veeramani P:
**Existence and convergence of best proximity points.***Journal of Mathematical Analysis and Applications*2006,**323**(2):1001–1006. 10.1016/j.jmaa.2005.10.081MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.