# Nonlinear conditions for the existence of best proximity points

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Research

## Abstract

In this paper, we first introduce the new notion of -cyclic contraction and establish some new existence and convergence theorems of iterates of best proximity points for -cyclic contractions. Some nontrivial examples illustrating our results are also given.

MSC:41A17, 47H09.

### Keywords

cyclic map best proximity point -function (-function) -cyclic contraction

## 1 Introduction and preliminaries

Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. Let A and B be nonempty subsets of a nonempty set E. A map is called a cyclic map if and . Let be a metric space and be a cyclic map. For any nonempty subsets A and B of X, let

A point is called to be a best proximity point for T if .

Definition 1.1 ([1])

Let A and B be nonempty subsets of a metric space . A map is called a cyclic contraction if the following conditions hold:
1. (1)

and ;

2. (2)

there exists such that for all , .

Remark 1.1 Let A and B be nonempty closed subsets of a complete metric space and be a cyclic contraction. If , then and T is a contraction on the complete metric space . Hence, applying the Banach contraction principle, we know that T has a unique fixed point in .

Recently, under some weaker assumptions over a map T, the existence, uniqueness and convergence of iterates to the best proximity point were investigated by several authors; see [1, 2, 3, 4, 5, 6] and references therein. In [1], Eldred and Veeramani first proved the following interesting best proximity point theorem.

Theorem EV ([[1], Proposition 3.2])

LetAandBbe nonempty closed subsets of a complete metric spaceX. Letbe a cyclic contraction map, and define, . Supposehas a convergent subsequence inA. Then there existssuch that.

Let f be a real-valued function defined on . For , we recall that
and

Definition 1.2 ([7, 8, 9, 10, 11, 12, 13])

A function is said to be an -function (or -function) if for all .

It is obvious that if is a nondecreasing function or a nonincreasing function, then φ is an -function. So the set of -functions is a rich class.

Very recently, Du [10] first proved some characterizations of -functions.

Theorem D ([10])

Letbe a function. Then the following statements are equivalent.
1. (a)

φis an-function.

2. (b)

For each, there existandsuch thatfor all.

3. (c)

For each, there existandsuch thatfor all.

4. (d)

For each, there existandsuch thatfor all.

5. (e)

For each, there existandsuch thatfor all.

6. (f)

For any nonincreasing sequencein, we have.

7. (g)

φis a function of contractive factor [8]; that is, for any strictly decreasing sequencein, we have.

Motivated by the concepts of cyclic contractions and -functions, we first introduce the concept of -cyclic contractions.

Definition 1.3 Let A and B be nonempty subsets of a metric space . If a map satisfies

(MT1) and ;

(MT2) there exists an -function such that

then T is called an -cyclic contraction with respect toφ on .

Remark 1.2 It is obvious that (MT2) implies that T satisfies for any and .

The following example gives a map T which is an -cyclic contraction but not a cyclic contraction.

Example A Let be a countable set and be a strictly increasing convergent sequence of positive real numbers. Denote by . Then . Let be defined by for all and if . Then d is a metric on X. Set , . So and . Now we define a map by
for . It is easy to see that and , and so (MT1) holds. Define as
Since is strictly increasing, for all . Hence φ is an -function. Clearly, and . For with and , . So . We claim that T is not a cyclic contraction on . Indeed, suppose that T is a cyclic contraction on . Thus there exists such that

From (∗), we get , which is a contradiction. Therefore, T is not a cyclic contraction on .

Next, we show that T is an -cyclic contraction with respect to φ. To verify (MT2), we need to observe the following cases:
1. (i)
Since , we obtain

2. (ii)
For with and , we have

From above, we can prove that (MT2) holds. Hence T is an -cyclic contraction with respect to φ. Moreover, since , is a best proximity point for T.

In this paper, we establish some new existence and convergence theorems of iterates of best proximity points for -cyclic contractions. Our results include some known results in the literature as special cases.

## 2 Existence and convergence theorems for best proximity points

First, we establish the following convergence theorem for -cyclic contractions, which is one of the main results in this paper.

Theorem 2.1LetAandBbe nonempty subsets of a metric spaceandbe an-cyclic contraction with respect toφ. Then there exists a sequencesuch that.

Proof Let be given. Define an iterative sequence by for . Clearly, for all . If there exists such that , then . So it suffices to consider the case for all . By Remark 1.2, it is easy to see that the sequence is nonincreasing in . Then
(2.1)
Since φ is an -function, applying Theorem D, we get
Let . Then for all . If , then, by (MT1), we have and for all . Notice first that (MT2) implies that
and
On the other hand, if , then and for all . Applying (MT2) again, we also have
and
Hence, by induction, one can obtain
(2.2)

Since , . Using (2.1) and (2.2), we obtain . The proof is completed. □

Here we give a nontrivial example illustrating Theorem 2.1.

Example B Let be a countable set. Define a strictly decreasing sequence of positive real numbers by for all . Then . Let be defined by for all and if . Then d is a metric on X. Set and . So
Let be defined by
It is easy to see that and and so (MT1) holds. Let be given. Define an iterative sequence by for . So and are identical, and hence . Define as
Since for all , φ is an -function. Now, we verify (MT2). For with , since is strictly decreasing and ,

which prove that (MT2) holds. Hence T is an -cyclic contraction with respect to φ. Therefore, all the assumptions of Theorem 2.1 are satisfied and the conclusion can follow from Theorem 2.1.

The following best proximity point theorem can be given immediately from Theorem 2.1.

Theorem 2.2LetAandBbe nonempty subsets of a metric spaceandbe a cyclic map. Suppose that there exists a nondecreasing (or nonincreasing) functionsuch that

Then there exists a sequencesuch that.

Corollary 2.1 ([1])

LetAandBbe nonempty subsets of a metric spaceandbe a cyclic contraction. Then there exists a sequencesuch that.

Here, we give an existence theorem for best proximity points.

Theorem 2.3LetAandBbe nonempty subsets of a metric spaceandbe a cyclic map. Letbe given. Define an iterative sequencebyfor. Suppose that
1. (i)

for anyand;

2. (ii)

has a convergent subsequence inA;

3. (iii)

.

Then there existssuch that.

Proof Since T is a cyclic map and , and for all . By (ii), has a convergent subsequence and as for some . Since
it follows from and the condition (iii) that . By (i), we have

which implies . □

Applying Theorems 2.1 and 2.3, we establish the following new best proximity point theorem for -cyclic contractions.

Theorem 2.4LetAandBbe nonempty subsets of a metric spaceandbe an-cyclic contraction with respect toφ. Letbe given. Define an iterative sequencebyfor. Suppose thathas a convergent subsequence inA, then there existssuch that.

Remark 2.1 ([[1], Proposition 3.2])

(i.e. Theorem EV) is a special case of Theorem 2.4.

## Notes

### Acknowledgements

The first author was supported partially by the grant No. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.

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