Unique Coincidence and Fixed Point Theorem for -Weakly C-Contractive Mappings in Partial Metric Spaces

We prove that every map satisfying the -weakly C-contractive inequality in partial metric space has a unique coincidence point. Our results generalize several well-known existing results in the literature. 1. Introduction and Preliminaries The Banach contraction principle is the source of metric fixed point theory. This principle had been extended by many authors in different directions (see [1]). Chatterjea [2] introduced the following contraction which has been named later as C-contraction. Definition 1 (see [2]). Let be a metric space and a mapping. Then is called a C-contraction if there exists such that holds for all , . Under this kind of contractive inequality, Chatterjea [2] established the following fixed point result. Theorem 2 (see [2]). Every C-contraction in a complete metric space has a unique fixed point. As a generalization of C-contractive mapping, Choudhury [3] introduced the concept of weakly C-contractive mapping and proved that every weakly C-contractive mapping in a complete metric space has a unique fixed point. Definition 3 (see [3]). Let be a metric space and a mapping. Then is called a weakly C-contractive if satisfies for all , , where is a continuous mapping such that if and only if . Under this kind of contraction, Choudhury [3] established the following fixed point result. Theorem 4 (see [3, Theorem 2.1]). Every weakly C-contraction in a complete metric space has a unique fixed point. Recently, Harjani et al. [4] studied some fixed point results for weakly C-contractive mappings in a complete metric space endowed with a partial order. Moreover, Shatanawi [5] proved some fixed point and coupled fixed point theorems for a nonlinear weakly C-contraction type mapping in metric and ordered metric spaces. In another aspect, the notion of a partial metric space has been introduced by Matthews [6] in 1994 as a generalization of the usual metric in such a way that each object does not necessarily have to have a zero distance from itself. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle (see, e.g., [7, 8]). Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions on partial metric spaces (e.g., [9–13]). We recall some definitions and properties of partial metric spaces. Definition 5. A partial metric on a nonempty set is a function such that for all , , , (p1) ; (p2) ;