Open Access Research

Coupled fixed point theorems for nonlinear contractions without mixed monotone property

Wutiphol Sintunavarat1, Poom Kumam1* and Yeol J Cho2*

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand

2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea

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Fixed Point Theory and Applications 2012, 2012:170 doi:10.1186/1687-1812-2012-170

Published: 3 October 2012

Abstract

In this paper, we show the existence of a coupled fixed point theorem of nonlinear contraction mappings in complete metric spaces without the mixed monotone property and give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled fixed point by using the mixed monotone property. We also study the necessary condition for the uniqueness of a coupled fixed point of the given mapping. Further, we apply our results to the existence of a coupled fixed point of the given mapping in partially ordered metric spaces. Moreover, some applications to integral equations are presented.

MSC: 47H10, 54H25.

Keywords:
coupled fixed point; F-invariant set; transitive property; mixed monotone property; partially ordered set