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
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
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 .
Lemma 2.3 (see ).
We use the following lemmas proved in .
3. Main Results
It is an interesting fact to note that the distances between the adjacent sets are equal under the p-cyclic nonexpansive mapping.
Hence, every p-cyclic Meir-Keeler contraction is a p-cyclic nonexpansive map.
Similarly, (2) can easily be proved.
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.
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 .
The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.
- Meir A, Keeler E: A theorem on contraction mappings. Journal of Mathematical Analysis and Applications 1969,28(2):326–329. 10.1016/0022-247X(69)90031-6MathSciNetView ArticleMATHGoogle Scholar
- Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003,4(1):79–89.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
- Di Bari C, Suzuki T, Vetro C: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):3790–3794. 10.1016/j.na.2007.10.014MathSciNetView ArticleMATHGoogle Scholar
- Amini-Harandi A, Farajzadeh AP, O'Regan D, Agarwal RP: Coincidence point, best approximation and best proximity theorems for condensing set-valued maps in hyperconvex metric spaces. Fixed Point Theory and Applications, acceptedGoogle Scholar
- Amini-Harandi A, Farajzadeh AP, O'Regan D, Agarwal RP: Best proximity pairs for upper semicontinuous set-valued maps in hyperconvex metric spaces. Fixed Point Theory and Applications 2008, Article ID 648985, 2008:-5.Google Scholar
- Lim T-C: On characterizations of Meir-Keeler contractive maps. Nonlinear Analysis: Theory, Methods & Applications 2001,46(1):113–120. 10.1016/S0362-546X(99)00448-4MathSciNetView ArticleMATHGoogle Scholar
- Suzuki T: Some notes on Meir-Keeler contractions and -functions. Bulletin of the Kyushu Institute of Technology 2006, (53):1–13.Google Scholar
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