Weakly Compatible Mappings and Common Fixed Points Under Generalized Contractive Conditions

TL;DR

This paper introduces generalized fixed point theorems for weakly compatible mappings in metric spaces, relaxing traditional conditions and broadening applicability beyond previous results.

Contribution

It develops a unified framework for common fixed points under generalized contractive conditions, removing the need for continuity and compatibility assumptions.

Findings

Generalizes existing fixed point theorems

Eliminates the need for space completeness

Includes examples demonstrating the necessity of hypotheses

Abstract

This paper establishes new common fixed point theorems for weakly compatible mappings in metric spaces, relaxing traditional requirements such as continuity, compatibility, and reciprocal continuity. We present a unified framework for three self-mappings $T$, $f$, and $g$ with a contractive condition involving a control function $\psi$, along with corollaries extending results to pairs of mappings and upper semi-continuous control functions. Further generalizations include iterated mappings and sequences of mappings. Rigorous examples demonstrate the necessity of hypotheses and show our results strictly generalize theorems by Al-Thagafi \emph{et. al.} \cite{Al-Thagafi2006}, Babu \emph{et. al.} \cite{Babu2007}, Jungck \cite{Jungck1976,Jungck1986}, Singh \cite{Singh1986,Singh1997a}, Som \cite{Som2003}, Song \cite{Song2007} and Zhang \emph{et. al.} \cite{Zhang2008}. Key advancements include eliminating completeness assumptions on the entire space and relaxing mapping compatibility conditions.