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RESEARCH

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Existence and uniqueness of fixed point for mixed monotone ternary operators with application Changchang Bu1,2 , Yuqiang Feng1,2* and Hui Li1,2 * Correspondence: [email protected] 1 School of Science, Wuhan University of Science and Technology, Wuhan, 430065, P.R. China 2 Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan, 430065, P.R. China

Abstract In this paper, partial order theory is used to study the fixed point of a mixed monotone ternary operator A : P × P × P → P. The existence and uniqueness of a fixed point are obtained without assuming the operators to be compact or continuous. In the end, the application to an integral equation is presented. Our results unify, generalize, and complement various known comparable results from the current literature. MSC: 47H05; 47H10 Keywords: fixed point; mixed monotone ternary operator; normal cone

1 Introduction Fixed point theory has fascinated hundreds of researchers since  with the celebrated Banach fixed point theorem. It is well known that mixed monotone operators were introduced by Guo and Lakshmikantham [] in . Later, Bhaskar and Lakshmikantham [] introduced the notion of a coupled fixed point and proved some coupled fixed point results under certain conditions, in a complete metric space endowed with a partial order. Their study has not only important theoretical meaning but also wide applications in engineering, nuclear physics, biological chemistry technology, etc. (see [–] and the references therein). Very recently, Harjani et al. [] have established the existence results of coupled fixed point for mixed monotone operators, and further obtained their applications to integral equations. Berinde and Borcut [] have introduced the concept of a triple fixed point and proved some related theorems for contractive type operators in partially ordered metric spaces. Zhai [] has considered mixed monotone operators with convexity and get the existence and uniqueness of a fixed point (A(u, u) = u type) without assuming the operator to be compact or continuous. Motivated by the work reported in [–], the aim of this paper is to discuss the existence and uniqueness of a fixed point (A(u, u, u) = u type) for mixed monotone ternary operators in the context of ordered metric spaces. Our results unify, generalize, and complement various known comparable results from the current literature. The rest of the paper is organized as follows. In Section , we recall some basic definitions and notations which will be used in the sequel. The existence and uniqueness of ©2014 Bu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Bu et al. Fixed Point Theory and Applications 2014, 2014:223 http://www.fixedpointtheoryandapplications.com/content/2014/1/223

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