Coupled fixed point theorems for $\phi$-contractive mixed monotone mappings in partially ordered metric spaces

TL;DR

This paper generalizes coupled fixed point theorems for mixed monotone operators in partially ordered metric spaces by weakening the contractive conditions, with applications to nonlinear Fredholm integral equations.

Contribution

It extends existing fixed point theorems by relaxing contractive conditions, broadening their applicability in nonlinear analysis.

Findings

Generalized fixed point theorems under weaker conditions

Provided an example illustrating the theorems

Applied results to nonlinear Fredholm integral equations

Abstract

In this paper we extend the coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ obtained in [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. \textbf{65} (2006) 1379-1393] and [N.V. Luong and N.X. Thuan, \textit{Coupled fixed points in partially ordered metric spaces and application}, Nonlinear Anal. \textbf{74} (2011) 983-992], by weakening the involved contractive condition. An example as well an application to nonlinear Fredholm integral equations are also given in order to illustrate the effectiveness of our generalizations.