Common fixed points of generalized contractions on partial metric spaces and an application

Keywords : Coincidence point Common fixed point Partial metric space Weakly compatible pair of mappings Homotopy a b s t r a c t In this paper , common fixed point theorems for four mappings satisfying a generalized nonlinear contraction type condition on partial metric spaces are proved. Presented theorems extend the very recent results of I. Altun , F. Sola and H. Simsek [ Generalized contractions on partial metric spaces , Topology and its applications 157 (18) (2010) 2778 – 2785 ]. As application , some homotopy results for operators on a set endowed with a partial metric are given. Partial metric spaces were introduced by Matthews [ 8 ] in 1992 as a part of the study of denotational semantics of data-flow networks. In fact , it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation [ 6 , 10 , 14 – 16 , 19 ]. First , we recall some definitions of partial metric space and some their properties [ 8 – 10 , 13 , 18 ]. A partial metric space is a pair (X , p) such that X is a nonempty set and p is a partial metric on X. Remark 1. 1. It is clear that , if p (x , y) = 0 , then from (p1) and (p2) , x = y. But if x = y , p (x , y) may not be 0. A basic example of a partial metric space is the pair ðR þ ; pÞ , where p (x , y) = max{x , y} for all x ; y 2 R þ. Each partial metric p on X generates a T 0 topology s p on Xwhich has as a base the family of open p-balls {B p