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 Applied Mathematics (AM) , 2015, DOI: 10.4236/am.2015.68130 Abstract: In this paper, first we introduce notions of (α, Ψ)-contractive and (α)-admissible for a pair of map and prove a coupled coincidence point theorem for compatible mappings using these notions. Our work extends and generalizes the results of Mursaleen et al. [1]. At the end, we will provide an example in support of our result.
 Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/701910 Abstract: The aim of this paper is to present some coincidence and common fixed point results for generalized weakly -contractive mappings in the setup of partially ordered -metric space. We also provide an example to illustrate the results presented herein. As an application of our results, periodic points of weakly -contractive mappings are obtained.
 Applied Mathematics (AM) , 2014, DOI: 10.4236/am.2014.521321 Abstract: In this paper, we use the mappings with quasi-contractive conditions, defined on a partially ordered set with cone metric structure, to construct convergent sequences and prove that the limits of the constructed sequences are the unique (common) fixed point of the mappings, and give their corollaries. The obtained results improve and generalize the corresponding conclusions in references.
 Fixed Point Theory and Applications , 2011, Abstract: Using the notion of compatible mappings in the setting of a partially ordered metric space, we prove the existence and uniqueness of coupled coincidence points involving a ( , ψ)-contractive condition for a mappings having the mixed g-monotone property. We illustrate our results with the help of an example.
 International Journal of Analysis , 2014, DOI: 10.1155/2014/586096 Abstract: The object of this paper is to establish the existence and uniqueness of coupled fixed points under a ( , )-contractive condition for mixed monotone operators in the setup of partially ordered metric spaces. Presented work generalizes the recent results of Berinde (2011, 2012) and weakens the contractive conditions involved in the well-known results of Bhaskar and Lakshmikantham (2006), and Luong and Thuan (2011). The effectiveness of our work is validated with the help of a suitable example. As an application, we give a result of existence and uniqueness for the solutions of a class of nonlinear integral equations. 1. Introduction and Preliminaries Fixed point theory is an important tool for studying the phenomenon of nonlinear analysis and is a bridge bond between pure and applied mathematics. The theory has its wide applications in engineering, computer science, physical and life sciences, economics, and other fields. Banach [1] introduced the well-known classical and valuable theorem in nonlinear analysis, which is named after him, known as the Banach contraction principle. This celebrated principle has been extended and improved by various authors in many ways over the years (see for instance [2–17]). Nowadays, fixed point theory has been receiving much attention in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering. Ran and Reurings [17] were the first to establish the results in this direction. The results were then extended by Nieto and Rodríguez-López [10] for nondecreasing mappings. Works noted in [18–24] are some examples in this direction. The work of Bhaskar and Lakshmikantham [25] is worth mentioning, as they introduced the new notion of fixed points for the mappings having domain the product space , which they called coupled fixed points, and thereby proved some coupled fixed point theorems for mappings satisfying the mixed monotone property in partially ordered metric spaces. As an application, they discussed the existence and uniqueness of a solution for a periodic boundary value problem. Definition 1 (see [25]). Let be a partially ordered set. The mapping is said to have the mixed monotone property if is monotone nondecreasing in and monotone nonincreasing in ; that is, for any , , Definition 2 (see [25]). An element is called a coupled fixed point of the mapping if and . Bhaskar and Lakshmikantham [25] gave the following result. Theorem 3 (see [25]). Let be a partially ordered set and suppose there exists a metric on such that is a complete metric space. Let be a continuous mapping having the mixed
 International Journal of Mathematics and Mathematical Sciences , 2012, DOI: 10.1155/2012/313675 Abstract: We prove some fixed point theorems for a T-Hardy-Rogers contraction in the setting of partially ordered partial metric spaces. We apply our results to study periodic point problems for such mappings. We also provide examples to illustrate the results presented herein. 1. Introduction and Preliminaries The notion of a partial metric space was introduced by Matthews in [1]. In partial metric spaces, the distance of a point in the self may not be zero. After the definition of a partial metric space, Matthews proved the partial metric version of Banach fixed point theorem. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle, more suitable in this context [1]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions (e.g., [2–21], [22]). Existence of fixed points in partially ordered metric spaces has been initiated in 2004 by Ran and Reurings [23]. Subsequently, several interesting and valuable results have appeared in this direction [14]. The aim of this paper is to study the necessary conditions for existence of fixed point of mapping satisfying -Hardy-Rogers conditions in the framework of partially ordered partial metric spaces. Our results extend and strengthen various known results [8, 24]. In the sequel, the letters , , and will denote the set of real numbers, the set of nonnegative real numbers, and the set of nonnegative integer numbers, respectively. The usual order on (resp., on will be indistinctly denoted by or by . Consistent with [1, 8] (see [25–29]) the following definitions and results will be needed in the sequel. Definition 1.1 (see [1]). A partial metric on a nonempty set is a mapping such that for all , , , , . A partial metric space is a pair such that is a nonempty set and is a partial metric on . If , then and imply that . But converse does not hold always. A trivial example of a partial metric space is the pair , where is defined as . Each partial metric on generates a topology on which has as a base the family open -balls , where . On a partial metric space the concepts of convergence, Cauchy sequence, completeness, and continuity are defined as follows. Definition 1.2 (see [1]). Let be a partial metric space and let be a sequence in . Then (i) converges to a point if and only if (we may still write this as or ); (ii) is called a Cauchy sequence if there exists (and is
 Huseyin ISIK and Duran TURKOGLU Fixed Point Theory and Applications , 2013, DOI: 10.1186/1687-1812-2013-51 Abstract: In this article, we establish some fixed point theorems for weakly contractive mappings defined in ordered metric-like spaces. We provide an example and some applications in order to support the useability of our results. These results generalize some well-known results in the literature. We also derive some new fixed point results in ordered partial metric spaces.
 Vasile Berinde Mathematics , 2011, Abstract: In this paper we extend the coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ obtained in [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. \textbf{65} (2006) 1379-1393] and [N.V. Luong and N.X. Thuan, \textit{Coupled fixed points in partially ordered metric spaces and application}, Nonlinear Anal. \textbf{74} (2011) 983-992], by weakening the involved contractive condition. An example as well an application to nonlinear Fredholm integral equations are also given in order to illustrate the effectiveness of our generalizations.
 Abstract and Applied Analysis , 2010, DOI: 10.1155/2010/190701 Abstract: The purpose of this paper is to present a fixed point theorem using a contractive condition of rational type in the context of partially ordered metric spaces. 1. Introduction In [1], Jaggi proved the following fixed point theorem. Theorem 1.1. Let be a continuous selfmap defined on a complete metric space . Suppose that satisfies the following contractive condition: for all , , and for some with , then has a unique fixed point in . The aim of this paper is to give a version of Theorem 1.1 in partially ordered metric spaces. Existence of fixed point in partially ordered sets has been considered recently in [2–15]. Tarski's theorem is used in [7] to show the existence of solutions for fuzzy equations and in [9] to prove existence theorems for fuzzy differential equations. In [5, 6, 8, 11, 14], some applications to matrix equations and to ordinary differential equations are presented. In [3, 6, 16], it is proved that some fixed theorems for a mixed monotone mapping in a metric space endowed with a partial order and the authors apply their results to problems of existence and uniqueness of solutions for some boundary value problems. In the context of partially ordered metric spaces, the usual contractive condition is weakened but at the expense that the operator is monotone. The main idea in [8, 14] involves combining the ideas in the contraction principle with those in the monotone iterative technique [16]. 2. Main Result Definition 2.1. Let be a partially ordered set and . We say that is a nondecreasing mapping if for , . In the sequel, we prove the following theorem which is a version of Theorem 1.1 in the context of partially ordered metric spaces. Theorem 2.2. Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Let be a continuous and nondecreasing mapping such that with . If there exists with , then has a fixed point. Proof. If , then the proof is finished. Suppose that . Since is a nondecreasing mapping, we obtain by induction that Put . If there exists such that , then from , is a fixed point and the proof is finished. Suppose that for . Then, from (2.1) and as the elements and are comparable, we get, for , The last inequality gives us Again, using induction Put . Moreover, by the triangular inequality, we have, for , and this proves that as . So, is a Cauchy sequence and, since is a complete metric space, there exists such that . Further, the continuity of implies and this proves that is a fixed point. This finishes the proof. In what follows, we prove that Theorem 2.2 is still valid for ,
 Fixed Point Theory and Applications , 2010, Abstract: We introduce the concept of a -compatible mapping to obtain a coupled coincidence point and a coupled point of coincidence for nonlinear contractive mappings in partially ordered metric spaces equipped with -distances. Related coupled common fixed point theorems for such mappings are also proved. Our results generalize, extend, and unify several well-known comparable results in the literature.
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