%0 Journal Article
%T Some Fixed Point Theorems for Pre£¿i£¿-Hardy-Rogers Type Contractions in Metric Spaces
%A Satish Shukla
%A Stojan Radenovi£¿
%A Slavi£¿a Panteli£¿
%J Journal of Mathematics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/295093
%X We introduce some generalizations of Pre£¿i£¿ type contractions and establish some fixed point theorems for mappings satisfying Pre£¿i£¿-Hardy-Rogers type contractive conditions in metric spaces. Our results generalize and extend several known results in metric spaces. Some examples are included which illustrate the cases when new results can be applied while old ones cannot. 1. Introduction The well-known Banach contraction mapping principle states that if is a complete metric space and is a self-mapping such that for all , where , then there exists a unique such that . This point is called the fixed point of mapping . On the other hand, for mappings , Kannan [1] introduced the contractive condition: for all , where is a constant and proved a fixed point theorem using (2) instead of (1). The conditions (1) and (2) are independent, as it was shown by two examples in [2]. Reich [3], for mappings , generalized Banach and Kannan fixed point theorems, using contractive condition: for all , where are nonnegative constants with . An example in [3] shows that the condition (3) is a proper generalization of (1) and (2). For mapping Chatterjea [4] introduced the contractive condition: for all , where is a constant and proved a fixed point result using (4). £¿iri£¿ [5], for mappings , generalized all above mappings, using contractive condition: for all , where are nonnegative constants with . A mapping satisfying (5) is called Generalized contraction. Hardy and Rogers [6], for mappings , used the contractive condition: for all , where are nonnegative constants with and proved fixed point result. Note that condition (6) generalizes all the previous conditions. In 1965, Pre£¿i£¿ [7, 8] extended Banach contraction mapping principle to mappings defined on product spaces and 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 condition (7) in the case reduces to the well-known Banach contraction mapping principle. So, Theorem 1 is a generalization of the Banach fixed point theorem. Some generalizations and applications of Pre£¿i£¿ theorem can be seen in [9¨C18]. The -step iterative sequence given by (8) represents a nonlinear difference equation and the solution of this equation can be assumed to be a fixed point of ; that is, solution of (8) is a point such that . The Pre£¿i£¿
%U http://www.hindawi.com/journals/jmath/2013/295093/