%0 Journal Article
%T Best Proximity Point for Generalized Proximal Weak Contractions in Complete Metric Space
%A Erdal Karap£¿nar
%A V. Pragadeeswarar
%A M. Marudai
%J Journal of Applied Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/150941
%X We introduce a new class of nonself-mappings, generalized proximal weak contraction mappings, and prove the existence and uniqueness of best proximity point for such mappings in the context of complete metric spaces. Moreover, we state an algorithm to determine such an optimal approximate solution designed as a best proximity point. We establish also an example to illustrate our main results. Our result provides an extension of the related results in the literature. 1. Introduction and Preliminaries A self-mapping , defined on a metric space , is said to be a contraction if there exists a constant such that the inequality holds for all . Moreover, a self-mapping is called a contractive mapping if holds for all with . The celebrated Banach contraction principle says that if is complete, then every contraction has a unique fixed point. In fact, the fixed point of a contraction mapping is obtained as a limit of repeated iteration of the mapping for any (initial) point of . Let be the class of continuous, nondecreasing mapping such that is positive on and . A function is called an altering distance function. A mapping is called a weak- contraction if there exists a such that for each . The notion of weak- contraction was defined by Alber and Guerre-Delabriere [1] to generalize the well-known Banach contraction principle in the setting of Hilbert spaces. Later, Rhoades [2] noticed that most of the results of Alber and Guerre-Delabriere [1] are valid for any Banach space. Rhoades also proved the following generalization of the Banach contraction principle (see also [3¨C7]). Theorem 1. Let be a nonempty complete metric space and let be a weak- contraction on ; then has a unique fixed point. Recently, Dutta and Choudhury [8] proved the following generalization of Theorem 1 by using -weak contraction map. Theorem 2. Let be a nonempty complete metric space and let be a self-mapping satisfying the inequality for all , where of all function. Then has a unique fixed point. Let be the class of all function; £¿£¿ is a lower semicontinuous with if and only if . In [9] Dori£¿ proved the following generalization of Theorem 2 by using generalized -weak contractions which contains the -weak contractions as a subclass. Theorem 3. Let be a nonempty complete metric space and let be a generalized -weak contraction map; that is, satisfies the following inequality: where£¿£¿ ,£¿£¿ , and for all . Then has a unique fixed point. One of the aims of this paper is to extend Theorem 3 via best proximity point. For this purpose, we recollect the basic definitions and fundamentals results as
%U http://www.hindawi.com/journals/jam/2014/150941/