%0 Journal Article
%T A Generalization of Pre£¿i£¿ Type Mappings in Metric-Like Spaces
%A Satish Shukla
%A Brian Fisher
%J Journal of Operators
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/368501
%X We generalize the result of Pre£¿i£¿ in metric-like spaces by proving some common fixed point theorems for Pre£¿i£¿ type mappings in metric-like spaces. An example is given which shows that the generalization is proper. 1. Introduction and Preliminaries Let be any metric space, and let be any mapping; then is said to be a contraction on if there exists such that A point is called a fixed point of if . Banach [1] proved that every contraction on a complete metric space has a unique fixed point and this result is known as the Banach contraction principle. There are several generalizations of this famous principle. One such generalization is given by Pre£¿i£¿ [2, 3]. When studying the convergence of some particular sequences, Pre£¿i£¿ [2, 3] proved the following theorem. Theorem 1. Let be a complete metric space, a positive integer, and a mapping satisfying the following contractive type condition: for every , where are nonnegative constants such that . Then there exists a unique point such that . Moreover, if are arbitrary points in and for , then the sequence is convergent and . Note that the -step iterative sequence given by (3) represents a nonlinear difference equation. In view of Pre£¿i£¿ theorem, it is obvious that if this sequence is convergent (which is ensured by the Pre£¿i£¿ theorem) then the limit of the sequence is a fixed point of . The result of Pre£¿i£¿ is generalized by several authors, and some generalizations and applications of Pre£¿i£¿ theorem can be seen in [4¨C15]. On the other hand, Matthews [16] introduced the notion of a partial metric space as a part of the study of denotational semantics of a dataflow network. In this space, the usual metric is replaced by a partial metric with an interesting property that the self-distance of any point of space may not be zero. Further, Matthews showed that the Banach contraction principle is valid in a partial metric space and can be applied in program verifications. O'Neill [17] generalized the concept of a partial metric space a bit further by admitting negative distances. The partial metric defined by O'Neill is called the dualistic partial metric. Heckmann [18] generalized it by omitting the small self-distance axiom. The partial metric defined by Heckmann is called a weak partial metric. Recently, Amini-Harandi [19] generalized the partial metric spaces by introducing the metric-like spaces. Amini-Harandi introduced the notion of a -Cauchy sequence and completeness of metric-like spaces and proved some fixed point theorems in such spaces. In this paper, we prove some common fixed point theorems for Pre£¿i£¿
%U http://www.hindawi.com/journals/joper/2013/368501/