Abstract:
We obtain coupled coincidence and coupled common fixed point theorems for mixed $g$-monotone nonlinear operators $F:X \times X \rightarrow X$ in partially ordered metric spaces. Our results are generalizations of recent coincidence point theorems due to Lakshmikantham and \' Ciri\' c [Lakshmikantham, V., \' Ciri\' c, L., \textit{Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces}, Nonlinear Anal. \textbf{70} (2009), 4341-4349], of coupled fixed point theorems established by Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. \textbf{65} (2006) 1379-1393] and also include as particular cases several related results in very recent literature.

Abstract:
In this paper coupled coincidence points of mappings satisfying a nonlinear contractive condition in the framework of partially ordered metric spaces are obtained. Our results extend the results of Harjani et al. (2011). Moreover, an example of the main result is given. Finally, some coupled coincidence point results for mappings satisfying some contraction conditions of integral type in partially ordered complete metric spaces are deduced.

Abstract:
Using the concept of a mixed g-monotone mapping, we prove some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete quasi-metric spaces with a Q-function q. The presented theorems are generalizations of the recent coupled fixed point theorems due to Bhaskar and Lakshmikantham (2006), Lakshmikantham and iri (2009) and many others.

Abstract:
The present study introduces the notion of compatibility in partially ordered G-metric spaces and uses this perception to establish a coupled coincidence point result. Our effort extend the recent work of Choudhary and Maity [B. S. Choudhary, P. Maity, Coupled fixed point results in generalized metric spaces, Mathematical and Computer Modelling 54 (2011) 73-79]. The example demonstrates that our main result is an actual improvement over the results which are generalized

Abstract:
We introduce and study new types of mixed monotone multivalued mappings in partially ordered complete metric spaces. We give relationships between those two types of mappings and prove their coupled fixed point and coupled common fixed point theorems in partially ordered complete metric spaces. Some examples of each type of mappings satisfying the conditions of the main theorems are also given. Our main result includes several recent developments in fixed point theory of mixed monotone multivalued mappings. 1. Introduction and Preliminaries Let be a metric space, and let be the class of all nonempty bounded and closed subsets of . For and , we denote . For , define If , then we write . Also in addition, if , then . For all , the definition of gives the following formulas: (i) ;(ii) ;(iii) if and only if ;(iv) . It is easy to see that the following inequality holds, for , for all . By using the above inequality, the following lemma is obtained. Lemma 1. Let be a metric space, and let be a nonempty subset of . If is defined by then is continuous. Let be a nonempty set, (collection of all nonempty subsets of ), and . An element is called(i)coupled fixed point of if and ,(ii)coupled coincidence point of a hybrid pair if and ,(iii)coupled common fixed point of a hybrid pair if and . We denote the set of coupled coincidence point of mappings and by . Note that if , then is also in . The hybrid pair is called -compatible if whenever . The mapping is called -weakly commuting at some point if and . Let be a partially ordered set, and suppose that there is a metric on such that is a metric space. For , we write if and . On the product space , we consider the following partial order: The existence of a fixed point for contraction type of mappings in partially ordered metric spaces has been considered recently by Ran and Reurings [1], Gnana Bhaskar and Lakshmikantham [2], Nieto and Rodríguez-López [3, 4], Agarwal et al. [5], Lakshmikantham and ？iri？ [6], and Harjani and Sadarangani [7]. In [2], Gnana Bhaskar and Lakshmikantham introduced the notions of mixed monotone mappings and a coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings. In [3], Nieto and Rodríguez-López studied some fixed point theorems for monotone nondecreasing mappings in partially ordered metric spaces. They proved the existence of a fixed point in partially ordered metric spaces and applied the obtained results to study a problem of ordinary differential equations. In [6], Lakshmikantham and ？iri？ introduced notions of a mixed -monotone mapping and proved

Abstract:
We prove a coupled coincidence point theorem for mappings F : ×→ andg : →, where F has the mixed g-monotone property, in partially ordered metric spaces viaimplicit relations. Our result extends and improves several results in the literature. Examples arealso given to illustrate our work.

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.

Abstract:
Cho et al. (2012) proved some coupled fixed point theorems in partially ordered cone metric spaces by using the concept of a c-distance in cone metric spaces. In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of a c-distance. Our results generalize several well-known comparable results in the literature. Also, we introduce an example to support the usability of our results.

Abstract:
Tripled fixed points are extensions of the idea of coupled fixed points introduced in a recent paper by Berinde and Borcut, 2011. Here using a separate methodology we extend this result to a triple coincidence point theorem in partially ordered metric spaces. We have defined several concepts pertaining to our results. The main results have several corollaries and an illustrative example. The example shows that the extension proved here is actual and also the main theorem properly contains all its corollaries.

Abstract:
Coupled fixed point theorems for a map satisfying mixed monotone property and a nonlinear, rational type contractive condition are established in a partially ordered -metric space. The conditions for uniqueness of the coupled fixed point are discussed. We also present results for the existence of coupled coincidence points of two maps. 1. Introduction The idea of weakening the contractive condition in a metric space by introducing partial order in the space and considering monotone functions satisfying contractive conditions was first developed by Ran and Reurings [1]. Later, this was extended by Bhaskar and Lakshmikantham [2] to prove a coupled fixed point theorem for functions satisfying mixed monotone property. Since then, there has been considerable interest in the development of coupled fixed point theorems in partially ordered metric spaces with a variety of contractive conditions [3–18]. Nonlinear contractive conditions were considered in [4, 6, 19]. In particular, a rational type contractive condition was considered by Jaggi [19] in a complete metric space and this was extended to a partially ordered complete metric space by Harjani et al. [6] to prove some fixed point theorems. Some coupled fixed point theorems in partially ordered, complete metric spaces were developed by Choudhury and Maity [8] and Saadati et al. [9]. The contractive conditions used in [8] were extensions of that used by Bhaskar and Lakshmikantham [2] into a metric space. A new concept of an distance was introduced in [9]. In this paper we develop a coupled fixed point theorem using a rational type, nonlinear contractive condition in a partially ordered complete metric space. The condition is similar to the rational type contractive condition of iri et al. [3] and may be considered as a generalization of the condition given in [3]. We also find conditions for the uniqueness of the coupled fixed point. Finally we consider the conditions for existence of coupled coincidence points. We begin by introducing the basic definitions and notions used in the paper. 2. General Preliminaries Throughout this work will denote a partial order relation on some given set. For any two elements in some partially ordered set endowed with the partial order relation , and are equivalent. Also by we mean and . Definition 1 (see [20]). Let be a nonempty set and let ？？: be a function satisfying the following properties:(1) if ,(2) for all with ,(3) for all with ,(4) (symmetry in all three variables),(5) for all (rectangle inequality). Then is called a generalized metric or more specifically a metric