Common coupled coincidence and coupled fixed points in $G$-metric spaces

Bhaskar and Lakshimkantham [T. G. Bhashkar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393] proved the existence of a coupled fixed point for a single valued mapping under weak contractive conditions and as an application they proved the existence of a unique solution of a boundary value problem associated with a first order ordinary differential equation. Lakshmikantham and iri [V. Lakshmikantham and Lj. iri , Coupled fixed point theorems for nonlinear contractions in partially ordered metric space, Nonlinear Anal. 70 (2009) 4341-4349] obtained a coupled coincidence and a coupled common fixed point of two single valued maps. In this paper, using the setting of a partially ordered generalized metric space, common coupled coincidence point results are obtained for hybrid pair of two mappings without exploiting the notion of continuity of mappings involved herein. As an application, we obtain a coupled fixed point of mapping in such spaces. We give an example of concepts introduced herein. These results generalize several well known comparable results in the literature.