Common fixed point of nonlinear contractive mappings

The purpose of this paper is to study the existence of a common fixed point for a pair of mappings without assumption of the contractive coefficient being fixed and less than 1. By replacing the fixed contractive coefficient with a nonlinear contractive function, we establish a unique common fixed point theorem for a pair of asymptotically regular self-mappings with either orbital continuity or $q$-continuity in a metric space. Moreover, by the asymptotical regularity of two approximate mappings, we prove that a pair of nonexpansive and continuous self-mappings, which are defined on a nonempty closed convex subset of a Banach space, have a common fixed point. Some examples are given to illustrate that our results are extensions of a recent result in the existing literature.