Common Fixed Point Theorems for Mappings under (CLRS)-Property in Partial Metric Spaces

Abstract

Abstract. In this paper, we introduce the concept of (CLRS)-property for mappings F : XˆX Ñ X and S : X Ñ X (wherein X stands for a partial metric space) and utilize the same to prove two common fixed point theorems for two pairs of mappings in partial metric spaces. We also furnish two examples to illustrate our main theorems. Introduction with preliminariesIn 2002, Aamri and El-Moutawakil [1] introduced the idea of the property (E.A) for a pair of self mappings defined on a metric space, which contains the classes of non-vacuously compatible as well as non-compatible mappings in metric spaces as proper subclasses and utilized the same to prove common fixed point theorems under strict contractive condition. Although the property (E.A) is a generalization of the concepts of non-compatible as well as non-vacuous compatible pairs, yet such results do require either completeness of the whole space or any one of the range subspaces or continuity conditions on the involved maps. But, quite contrary to this, the new notion of (CLR S )-property recently given by Sintunavarat and Kumam [8] does not impose such conditions. The importance of (CLR S )-property ensures that one does not require the completeness of the whole space or closedness of range subspaces.Nazir and Abbas [6] introduced the property (E.A) for a pair of self maps in partial metric spaces. In this paper, we introduce the concept of (CLR S )-property for the maps F : XˆX Ñ X and S : X Ñ X in a partial metric space and utilize the same to prove two unique common fixed point theorems for four mappings in partial metric spaces. As usual, let us denote 2010 Mathematics Subject Classification: 54H25, 47H10.

Common fixed point theorems for two mappings in b-metric-like spaces

“…Their fixed point results in b-metric-like spaces have generalized and improved some well-known results in the literature. Also, in [15,18,19] have given other fixed point results in b-metric-like spaces.…”