Coupled fixed point theorems for generalized symmetric Meir--Keeler contractions in ordered metric spaces

TL;DR

This paper introduces generalized symmetric Meir-Keeler contractions and proves new coupled fixed point theorems in ordered metric spaces, extending and unifying recent results in the field.

Contribution

It extends existing coupled fixed point theorems by introducing generalized contractions and provides a unifying framework for recent results in ordered metric spaces.

Findings

New coupled fixed point theorems established

Generalizations effectively unify previous results

An example demonstrates the applicability of the theorems

Abstract

In this paper we introduce generalized symmetric Meir-Keeler contractions and prove some coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ in partially ordered metric spaces. The obtained results extend, complement and unify some recent coupled fixed point theorems due to Samet [B. Samet, \textit{Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces}, Nonlinear Anal. \textbf{72} (2010), 4508-4517], Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. TMA \textbf{65} (2006) 1379-1393] and some other very recent papers. An example to show that our generalizations are effective is also presented.