Unified common fixed point theorems under weak reciprocal continuity or without continuity

Abstract

The purpose of this paper is two fold. Firstly, using the notion of weak reciprocal continuity due to Pant et al. Weak reciprocal continuity and fixed point theorems, Ann. Univ. Ferrara Sez. VII Sci. Mat. 57(1), 181-190 (2011)], we prove unified common fixed point theorems for various variants of compatible and $R$-weakly commuting mappings in complete metric spaces employing an implicit relation which covers a multitude of contraction conditions yielding thereby known as well as unknown results as corollaries. Secondly, we point out that more natural results can be proved under relatively tighter conditions if we replace the completeness of the space by completeness of suitable subspaces. The realized improvements in our results are also substantiated using appropriate examples.

Keywords

• metric space
• compatible mappings
• $R$-weakly commuting mappings
• weak reciprocal continuity
• coincidentally commuting
• implicit relation.