In this paper, our aim is to introduce the concept of G-g-contraction mapping and prove some coupled coincidence and coupled common fixed point theorems for nonlinear contraction mappings in the new set up of partially ordered complete metric spaces endowed with a directed graph. As an application, we apply our results to present an existence theorem for solution of some particular integral equations. Our paper is inspired by the work of Chifu and Petrusel (Fixed Point Theory Appl. 2014:151, 2014); the authors introduced the concept of a coupled fixed point. In the current paper, however, we have established the results by introducing the new notion of a coupled coincidence fixed point instead of the coupled fixed point in the setting of a partially ordered complete metric space with graph.
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机译：在本文中，我们的目的是引入G-g-压缩映射的概念，并在带有有向图的部分有序完整度量空间的新设置中证明非线性收缩映射的一些耦合重合和耦合公共不动点定理。作为一种应用，我们将我们的结果应用于提出一些特定积分方程解的存在性定理。本论文的灵感来自Chifu和Petrusel的工作（Fixed Point Theory Appl。2014：151，2014）;作者介绍了耦合不动点的概念。但是，在当前论文中，我们通过在图的部分有序完整度量空间的设置中引入耦合重合不动点的新概念而不是耦合不动点来建立结果。
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