Coupled coincidence point theorems for mixed (G, S)-monotone operators on partially ordered metric spaces and applications

TL;DR

This paper introduces new coupled coincidence point theorems for mixed (G, S)-monotone operators in partially ordered metric spaces, extending existing fixed point results and applying them to boundary value problems.

Contribution

It generalizes and improves recent fixed point theorems for mixed monotone operators by incorporating nonlinear contractions and altering distance functions.

Findings

Extended fixed point theorems for mixed (G, S)-monotone operators.

Proved coupled coincidence and coupled common fixed point theorems.

Applied results to periodic boundary value problems.

Abstract

In this paper, we introduce the concept of mixed (G, S)-monotone mappings and prove coupled coincidence and coupled common fixed point theorems for such mappings satisfying a nonlinear contraction involving altering distance functions. Presented theorems extend, improve and generalize the very recent results of Harjani, L\'opez and Sadarangani [J. Harjani, B. L\'opez and K. Sadarangani, Fixed point theorems for mixed monotone operators and applications to integral equations, Nonlinear Analysis (2010), doi:10.1016/j.na.2010.10.047] and other existing results in the literature. Some applications to periodic boundary value problems are also considered.