- Research Article
- Open Access

# Best Proximity Point Theorems for *p*-Cyclic Meir-Keeler Contractions

- S. Karpagam
^{1}Email author and - Sushama Agrawal
^{1}

**2009**:197308

https://doi.org/10.1155/2009/197308

© S. Karpagam and S. Agrawal. 2009

**Received:**31 August 2008**Accepted:**5 January 2009**Published:**19 January 2009

## Abstract

We consider a contraction map
of the Meir-Keeler type on the union of *p* subsets
,
, of a metric space (
) to itself. We give sufficient conditions for the existence and convergence of a best proximity point for such a map.

## Keywords

- Banach Space
- Point Theorem
- Nonexpansive Mapping
- Nonempty Subset
- Cauchy Sequence

## 1. Introduction

Meir and Keeler in [1] considered an extension of the classical Banach contraction theorem on a complete metric space. Kirk et al. in [2] extended the Banach contraction theorem for a class of mappings satisfying cyclical contractive conditions. Eldred and Veeramani in [3] introduced the following definition. Let and be nonempty subsets of a metric space . A map , is a cyclic contraction map if it satisfies

- (2)
for some .

In this case, a point
such that
, called a best proximity point, has been considered. This notion is more general in the sense that if the sets intersect, then every best proximity point is a fixed point. In [3], sufficient conditions for the existence and convergence of a unique best proximity point for a cyclic contraction on a uniformly convex Banach space have been given. Further, in [4], this result is extended by Di Bari et al., where the contraction condition of the map is of the Meir-Keeler-type. That is, in addition to the cyclic condition, if the map satisfies the condition that for a given
there exists a
such that
implies that
. Then, such a map is called a cyclic Meir-Keeler map. In [4], sufficient conditions are given to obtain a unique best proximity point for such maps. One may refer to [5, 6] for similar types of notion of best proximity points. A question that naturally arises is whether the main results in [4] can be extended to *p* subsets,
? From a geometrical point of view, for the cyclic Meir-Keeler contraction defined on the union of two sets, there is no question concerning the position of the sets. But in the case of more than two sets, the map is defined on the union of *p* sets,
(Definition 3.5), so that the image of
is contained in
and the image of
is contained in
but not in
(
and
). Hence, it is interesting to extend the notion of the cyclic Meir-Keeler contraction to *p* sets,
and we call this map a *p*-cyclic Meir-Keeler contraction. In this paper, we give sufficient conditions for the existence and convergence of a best proximity point for such a map (Theorem 3.13). Here, we observe that the distances between the adjacent sets are equal under this map, and this fact plays an important role in obtaining a best proximity point. Also, the obtained best proximity point is a periodic point of
with period *p*. Moreover, if
is a best proximity point in
, then
is a best proximity point in
for

## 2. Preliminaries

In this section, we give some basic definitions and concepts related to the main results. We begin with a definition due to Lim [7].

Definition 2.1.

A function
is called an *L*-function if
,
and for every
there exists
such that
for all
.

Let be a nonempty set, and let . Then, the following are equivalent.

(1)For each , there exists such that

(2)There exists an *L*-function
(nondecreasing, continuous) such that
and
.

Lemma 2.3 (see [8]).

Let
be an *L*-function. Let
be a nonincreasing sequence of nonnegative real numbers. Suppose
for all
with
, then,
.

It is well known that if
is a convex subset of a strictly convex normed linear space
and
, then a best approximation of *x* from
, if it exists, is unique.

We use the following lemmas proved in [3].

Lemma 2.4.

Let be a nonempty closed and convex subset and be a nonempty closed subset of a uniformly convex Banach space. Let be sequences in and let be a sequence in satisfying

(1) ,

(2)for every there exists such that for all .

Then, for every , there exists , such that for all .

Lemma 2.5.

Let be a nonempty closed and convex subsets and let be a nonempty closed subset of a uniformly convex Banach space. Let be sequences in and let be a sequence in satisfying

(1) ,

(2) .

Then, converges to zero.

## 3. Main Results

Definition 3.1.

A point is said to be a best proximity point if .

Definition 3.2.

*p*-cyclic mapping. is called a

*p*-cyclic nonexpansive mapping if

It is an interesting fact to note that the distances between the adjacent sets are equal under the *p*-cyclic nonexpansive mapping.

Lemma 3.3.

Let
be as in Definition 3.2. Then,
for all *i*,
.

Proof.

For , , , implies . That is, .

Remark 3.4.

If
is a best proximity point, then since
and since the distances between the adjacent sets are equal,
is a best proximity point of *T* in
for
.

Definition 3.5.

*p*-cyclic mapping. is called a

*p*-cyclic Meir-Keeler contraction if for every there exists such that

for all for .

Remark 3.6.

From Lemma 2.2, we see that
is a *p*-cyclic Meir-Keeler contraction if and only if there exists an *L*-function
(nondecreasing and continuous) such that for all
,
,
,
.

Remark 3.7.

From Remark 3.6, if
is a *p*-cyclic Meir-Keeler contraction, then for
,
,
, the following hold:

(1)

(2)

Hence, every *p*-cyclic Meir-Keeler contraction is a *p*-cyclic nonexpansive map.

Lemma 3.8.

Let be as in Definition 3.5, where each is closed. Then, for every for

(1)

(2)

Proof.

*n*, then for all . Since , we find that and this proves (1). Hence, assume for all

*n*. By Remark 3.7, , and by Remark 3.6, there exists an

*L*-function such that

Hence, . Therefore, .

Similarly, (2) can easily be proved.

Remark 3.9.

From Lemma 3.8, if is a uniformly convex Banach space and if each is convex, then for , . Then, by Lemma 2.5, . Similarly, .

Theorem 3.10.

Let
be as in Definition 3.5. If for some *i* and for some
, the sequence
in
contains a convergent subsequence
converging to
, then
is a best proximity point in
.

Proof.

Therefore, .

*p*-cyclic map which satisfies the following condition. For given there exists a such that

for all .

It follows from Lemma 2.2 that a *p*-cyclic map
satisfies the condition (3.6), if and only if there exists an *L*-function
(nondecreasing and continuous) such that for all
,
and for all *i*,
,
,
and
satisfies the *p*-cyclic nonexpansive property.

We use the following result due to Meir and Keeler [1] in the proof of Theorem 3.12.

Theorem 3.11.

Then, has a unique fixed point . Moreover, for any , the sequence converges to .

Theorem 3.12.

Let
be a complete metric space. Let
be nonempty closed subsets of
. Let
be a *p*-cyclic map satisfying (3.6). Then,
is nonempty and for any
,
, the sequence
converges to a unique fixed point in
.

Proof.

*n*, then by the

*p*-cyclic nonexpansive property of , . Therefore, assume for all

*n*. We note that the sequence is nonincreasing, and there exists an

*L*-function such that and by Lemma 2.3, . Now,

Fix
. By the definition of *L*-function, there exists
such that
.

Therefore,
. Since
for all *j*,
, and since
,
for all *i*,
. Therefore,
is a fixed point. Let
. Restricting
, we see that
is a Meir-Keeler contraction on the complete metric space
. Hence, by Theorem 3.11, *z* is the unique fixed point in
.

Now, we prove our main result.

Theorem 3.13.

Let
be nonempty, closed, and convex subsets of a uniformly convex Banach space. Let
be a *p*-cyclic Meir-Keeler contraction. Then, for each *i*,
, there exists a unique
such that for any
, the sequence
converges to
, which is a best proximity point in
. Moreover,
is a periodic point of period *p*, and
is a best proximity point in
for
.

Proof.

*i*, then for all

*i*, and hence, is nonempty. In this case, has a unique fixed point in the intersection. Therefore, assume for all

*i*. Let . There exists an

*L*-function as given in Remark 3.6. Fix . Choose satisfying . By Remark 3.9, . Hence, there exists such that

Thus, a contradiction. Hence, . Since and is convex, .

## Declarations

### Acknowledgment

The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.

## Authors’ Affiliations

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