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Unified Common Fixed Point Theorems for a Hybrid Pair of Mappings via an Implicit Relation Involving Altering Distance Function

DOI: 10.1155/2014/718040

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Abstract:

The object of this paper is to emphasize the role of a suitable implicit relation involving altering distance function which covers a multitude of contraction conditions in one go. By using this implicit relation, we prove a new coincidence and common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings in a metric space employing the common limit range property. Our main result improves and generalizes a host of previously known results. We also utilize suitable illustrative examples to substantiate the realized improvements in our results. 1. Introduction and Preliminaries Fixed point theory is one of the most rapidly growing research areas in nonlinear functional analysis. Apart from numerous extensions of Banach Contraction Principle for single valued mappings, it was also naturally extended to multivalued mappings by Nadler Jr. [1] in 1969 which is also sometimes referred to as Nadler Contraction Principle. Since then, there has been continuous and intense research activity in multimap fixed point theory (including hybrid fixed point results) and by now there exists an extensive literature on this specific theme (see, e.g., [2–7] and the references therein). The study of common fixed points of mappings satisfying hybrid contraction conditions has been at the center of vigorous research activity. Here, it can be pointed out that hybrid fixed theorems have numerous applications in science and engineering. In the following lines, we present some definitions and their implications which will be utilized throughout this paper. Let be a metric space. Then, on the lines of Nadler Jr. [1], we adopt that(1) is a nonempty closed subset of ,(2) is a nonempty closed and bounded subset of ,(3)for nonempty closed and bounded subsets of and , It is well known that is a metric space with the distance which is known as the Hausdorff-Pompeiu metric on . The following terminology is also standard. Let be a metric space with and . Then(1)a point is a fixed point of (resp., ) if (resp., ). The set of all fixed points of (resp., ) is denoted by (resp., );(2)a point is a coincidence point of and if . The set of all coincidence points of and is denoted by ;(3)a point is a common fixed point of and if . The set of all common fixed points of and is denoted by . In 1984, Khan et al. [8] utilized the idea of altering distance function in metric fixed point theory which is indeed a control function that alters distance between two points in a metric space. Thereafter, this idea has further been utilized by several mathematicians (see, e.g.,

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