Coupled fixed point theorems for rational inequality in generalized metric spaces (Q2799671)

The authors prove coupled fixed point theorems for continuous mappings having the mixed monotone property and satisfying a certain contractive condition in a \(G\)-metric space in the sense of \textit{Z. Mustafa} and \textit{B. Sims} [J. Nonlinear Convex Anal. 7, No. 2, 289--297 (2006; Zbl 1111.54025)].NEWLINENEWLINENote that Example 1 that is designated to illustrate one of the main results of the paper (Theorem 1) is a simple consequence of \textit{T. G. Bhaskar} and \textit{V. Lakshmikantham} original coupled fixed point theorems in usual metric spaces [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 65, No. 7, 1379--1393 (2006; Zbl 1106.47047)], based on the contractive type inequality NEWLINE\[NEWLINE \left|\frac{x-y}{16}-\frac{u-v}{16}\right|\leq k[|x-u|+|y-v|], NEWLINE\]NEWLINE which is satisfied for all \(x,y\in\mathbb R\) with the constant \(k=\frac{1}{8}<1\).