$(G,F)$- Closed Set and Coupled Coincidence Point Theorems for a Generalized Compatible in Partially Metric Spaces

Phakdi Charoensawan

Abstract

In this work, we prove the existence of a coupled coincidence point theorem for a pair $\{F,G\}$ of mapping $F,G:X\times X\to X$ with $\varphi$- contraction mappings in complete metric spaces without $G$-increasing property of $F$ and mixed monotone property of $G$ , using concept of $(G, F)$-closed set. We  give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled coincidence point by $G$ using the mixed monotone property. We also show  the uniqueness of a coupled coincidence point of the given mapping. Further, we apply our results to the existence and uniqueness of a coupled coincidence point of the given mapping in partially ordered metric spaces.