Coupled fixed point theorems for \((\varphi,\psi)\)-contractive mixed monotone mappings in partially ordered metric spaces and applications
Summary: The object of this paper is to establish the existence and uniqueness of coupled fixed points under a \((\varphi,\psi)\)-contractive condition for mixed monotone operators in the setup of partially ordered metric spaces. Presented work generalizes the recent results of \textit{V. Berinde} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 7347--7355 (2011; Zbl 1235.54024); ibid. 75, No. 6, 3218--3228 (2012; Zbl 1250.54043)] and weakens the contractive conditions involved in the well-known results of \textit{T. G. Bhaskar} and \textit{V. Lakshmikantham} [ibid. 65, No. 7, 1379--1393 (2006; Zbl 1106.47047)], and \textit{Nguyen Van Luong} and \textit{Nguyen Xuan Thuan} [ibid. 74, No. 3, 983--992 (2011; Zbl 1202.54036)]. The effectiveness of our work is validated with the help of a suitable example. As an application, we give a result of existence and uniqueness for the solutions of a class of nonlinear integral equations.
\((\varphi,\psi)\)-contractive condition
scientific article; zbMATH DE number 6883058