Coupled fixed point theorems in $G_{b}$-metric spaces

Abstract

T. G. Bhaskar and V. Lakshmikantham [Fixed point theoremsin partially ordered metric spaces and applications, Nonlinear Anal. 65(2006) 1379–1393], V. Lakshmikantham andLj\. B. Ćirić [Coupled fixed pointtheorems for nonlinear contractions in partially ordered metric spaces,Nonlinear Anal. 70 (2009) 4341–4349] introduced the concept of a coupledcoincidence point of a mapping $F$ from $X\times X$ into $X$ and a mapping$g$ from $X$ into $X$. In this paper we prove a coupled coincidence fixedpoint theorem in the setting of a generalized $b$-metric space. Threeexamples are presented to verify the effectiveness and applicability of ourmain result.