Best Proximity Point Theorems for p-Cyclic Meir-Keeler Contractions
© S. Karpagam and S. Agrawal. 2009
Received: 31 August 2008
Accepted: 5 January 2009
Published: 19 January 2009
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.
Meir and Keeler in  considered an extension of the classical Banach contraction theorem on a complete metric space. Kirk et al. in  extended the Banach contraction theorem for a class of mappings satisfying cyclical contractive conditions. Eldred and Veeramani in  introduced the following definition. Let and be nonempty subsets of a metric space . A map , is a cyclic contraction map if it satisfies
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 , 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 , 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 , 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  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
In this section, we give some basic definitions and concepts related to the main results. We begin with a definition due to Lim .
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 ).
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 .
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
(2)for every there exists such that for all .
Then, for every , there exists , such that for all .
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
Then, converges to zero.
3. Main Results
A point is said to be a best proximity point if .
It is an interesting fact to note that the distances between the adjacent sets are equal under the p-cyclic nonexpansive mapping.
Let be as in Definition 3.2. Then, for all i, .
For , , , implies . That is, .
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 .
for all for .
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 , , , .
From Remark 3.6, if is a p-cyclic Meir-Keeler contraction, then for , , , the following hold:
Hence, every p-cyclic Meir-Keeler contraction is a p-cyclic nonexpansive map.
Let be as in Definition 3.5, where each is closed. Then, for every for
Hence, . Therefore, .
Similarly, (2) can easily be proved.
From Lemma 3.8, if is a uniformly convex Banach space and if each is convex, then for , . Then, by Lemma 2.5, . Similarly, .
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 .
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  in the proof of Theorem 3.12.
Then, has a unique fixed point . Moreover, for any , the sequence converges to .
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 .
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.
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 .
Thus, a contradiction. Hence, . Since and is convex, .
The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.
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