Best Proximity Point Results in Complex Valued Metric Spaces

We introduce the concept of proximity points for nonself-mappings between two subsets of a complex valued metric space which is a recently introduced extension of metric spaces obtained by allowing the metric function to assume values from the field of complex numbers. We apply this concept to obtain the minimum distance between two subsets of the complex valued metric spaces. We treat the problem as that of finding the global optimal solution of a fixed point equation although the exact solution does not in general exist. We also define and use the concept of P-property in such spaces. Our results are illustrated with examples. 1. Introduction and Preliminaries In this paper we prove certain proximity point results to obtain the minimum distance between two subsets of a complex valued metric space. Essentially it is a global optimization problem which we treat here as the problem of finding the global optimal solution of a fixed point iteration. It is a part of the more general category of problems of finding minimum distances between two objects. In geometry it has led to the concept of geodesics, a curve along which the optimal distance between two given points of the space is realized [1]. Examples abound in physical theories, especially in the general theory of relativity, where finding the physically possible shortest path is sometimes the main task [2]. In proximity point problems our objects are sets. Here our aim is to find the distance between two sets and with the help of a function defined from to . Precisely we want to find a solution to the problem of minimizing the distance between and where is varied over the set . Equivalently we want to find the optimal solution of the equation although the exact solution does not in general exist as in the case where and are disjoint. It is at this point the best approximation theorems and the best proximity point theorems have their roles to play. The best approximation theorems provide the best approximate solutions which need not be globally optimal. For instance, let us consider the following Ky Fan’s best approximation theorem. Theorem 1 (see [3]). Let be a nonempty compact convex subset of a normed linear space and let be a continuous function. Then there exists such that . The element in the above theorem need not give the optimum value of . Proximity point result was first proved by [4]. After that several results on proximity have followed. Particularly, in the general setting of metric spaces there are a good number of results, and [5–14] are instances of these results. As we have already