Fixed point results for weakly compatible mappings in cone metric spaces and C*-algebra-valued metric spaces: Applications to functional equations

Abstract

This paper investigates fixed point theorems for weakly compatible and occasionally weakly compatible mappings in cone metric spaces and C*-algebra-valued metric spaces. We establish several common fixed-point results under various contractive conditions including rational expressions and integral-type contractions. The study provides a detailed analysis of weak commutativity conditions that are more general than strong commutativity. We prove existence and uniqueness theorems for pairs and families of mappings satisfying implicit relations in both normal and non-normal cone metric spaces. Applications to nonlinear functional equations, including those arising in dynamic programming and iterative functional equations, are presented with illustrative examples. Our results extend and unify several existing fixed point theorems in the literature while providing new insights into the structure of these generalized metric spaces.