Abstract
In this work, we present a notion of an -closed set and prove the existence of a coupled coincidence point theorem for a pair of mappings with φ-contraction mappings in partially ordered metric spaces without H-increasing property of F and mixed monotone property of H. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled coincidence point by H using the mixed monotone property and H-increasing property of F. We also show the uniqueness of a coupled coincidence point of the given mappings. Further, we apply our results to the existence and uniqueness of a coupled coincidence point of the given mappings in partially ordered G-metric spaces with H-increasing property of F and mixed monotone property of H. These results generalize some recent results in the literature.
Highlights
The existence of a fixed point for the contraction type of mappings in partially ordered metric spaces has been first studied by Ran and Reurings [ ]
In, Choudhury and Maity [ ] proved the existence of a coupled fixed point theorem of nonlinear contraction mappings with mixed monotone property in partially ordered G-metric spaces
Aydi et al [ ] established coupled coincidence and coupled common fixed point results for a mixed g-monotone mapping satisfying nonlinear contractions in partially ordered G-metric spaces
Summary
The existence of a fixed point for the contraction type of mappings in partially ordered metric spaces has been first studied by Ran and Reurings [ ]. They showed some applications on the existence and uniqueness of the coupled fixed point theorems for mappings which satisfy the mixed monotone property in partially ordered metric spaces.
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