Generalized coupled fixed point theorems for 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 contractive conditions, providing new proofs, examples, and applications to boundary value problems.

Contribution

It introduces significantly weaker contractive conditions for coupled fixed point theorems, with a novel proof technique and practical applications.

Findings

Extended fixed point theorems with weaker conditions

Provided an example illustrating the generalization

Applied results to periodic boundary value problems

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. TMA \textbf{65} (2006) 1379-1393] by significantly weakening the involved contractive condition. Our technique of proof is essentially different and more natural. An example as well an application to periodic BVP are also given in order to illustrate the effectiveness of our generalizations.