Altering Distance and Common Fixed Points under Implicit Relations

Abstract

We prove a common fixed point theorem for two pairs of occasionally weakly compatible mappings satisfying an implicit relation of a new type that involves an altering distance, so generalizing a theorem of Aliouche and Djoudi (Common fixed point theorems for mappings satisfying an implicit relation without decreasing assumption, Hacettepe J. Math. Stat. 36(1), 11–18, 2007). As a consequence, we obtain a fixed point theorem for two pairs of mappings satisfying an implicit relation of integral type, providing a strong generalization to a known result from Kumar, Chugh and Kumar (Fixed point theorems for compatible mappings satisfying a contractive condition of integral type, Soochow J. Math. 33(2), 181–185, 2007).