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

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.

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

(1) and and

• (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 .

Lemma 2.2 (see [7, 8]).

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.

Let be nonempty subsets of a metric space. Then, is called a p-cyclic mapping if

(3.1)

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

Definition 3.2.

Let be nonempty subsets of a metric space and be a p-cyclic mapping. is called a p-cyclic nonexpansive mapping if

(3.2)

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.

Let be nonempty subsets of a metric space . Let be a p-cyclic mapping. is called a p-cyclic Meir-Keeler contraction if for every there exists such that

(3.3)

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.

To prove (1), Lemma 2.3 is used. Let . If for some 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

(3.4)

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.

(3.5)

Therefore, .

Let be a metric space. Let be nonempty subsets of and let be a p-cyclic map which satisfies the following condition. For given there exists a such that

(3.6)

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.

Let be a complete metric space, and let be such that for given there exists a such that for all ,

(3.7)

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.

Let . Let . If for some 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,

(3.8)

Also, consider

(3.9)

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

Choose an satisfying

(3.10)

(3.11)

(3.12)

Let us show that

(3.13)

Let us do this by the method of induction. From (3.12), it is clear that (3.13) holds for . Fix . Assume that (3.7) is true for . Now,

(3.14)

By induction, (3.13) holds for all . Now, for all ,

(3.15)

Therefore, is a Cauchy sequence and converges to a point . Consider

(3.16)

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.

If for some 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

(3.17)

Let us prove that

(3.18)

Fix . It is clear that (3.18) is true for . Assume that (3.18) is true for . Now,

(3.19)

Hence, (3.18) holds for . Therefore, by induction, (3.18) is true for all . Note that . Now, by Lemma 2.4, for every , there exists such that for every , Hence, is a Cauchy sequence and converges to . By Theorem 3.10, z is a best proximity point in . That is, . Let such that and such that . Then, by Theorem 3.10, is a best proximity point. That is, . Let us show that . To do this,

(3.20)

Since is a convex set and is a uniformly convex Banach space, . Similarly, we can prove that . Now,

(3.21)

Since is convex, . Now, . If then there is nothing to prove. Therefore, let . This implies that

(3.22)

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