A coupled coincidence point result in partially ordered metric spaces for compatible mappings

In this paper we introduce the notion of compatibility of mappings in a partially ordered metric space and use this notion to establish a coupled coincidence point result. Our work extends the work of Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379-1393]. An example is also given.     

Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces

In this paper, we introduce the concept of tripled fixed point for nonlinear mappings in partially ordered complete metric spaces and obtain existence, and existence and uniqueness theorems for contractive type mappings. Our results generalize and extend recent coupled fixed point theorems established by Gnana Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379-1393]. Examples to support our new results are given.     

Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces

We introduce the concept of a mixed g-monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. Presented theorems are generalizations of the recent fixed point theorems due to Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393] and include several recent developments.     

Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces

Let X be a non-empty set and F:X×X→X be a given mapping. An element (x,y)∈X×X is said to be a coupled fixed point of the mapping F if F(x,y)=x and F(y,x)=y. In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393].     

A coupled best approximations theorem in normed spaces

We prove a coupled best approximations theorem in normed spaces. Also, we derive the results on coupled coincidence points and coupled fixed points, which were introduced by Lakshmikantham and ?iri? [V. Lakshmikantham, LJ. ?iri?, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. TMA, 70 (2009) 4341-4349].     

Multidimensional fixed point theorems in partially ordered complete metric spaces

In this paper we propose a notion of coincidence point between mappings in any number of variables and we prove some existence and uniqueness fixed point theorems for nonlinear mappings verifying different kinds of contractive conditions and defined on partially ordered metric spaces. These theorems extend and clarify very recent results that can be found in [T. Gnana-Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7)(2006) 1379–1393], [V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 4889–4897] and [M. Berzig, B. Samet, An extension of coupled fixed point’s concept in higher dimension and applications, Comput. Math. Appl. 63 (8) (2012) 1319–1334].     

Coupled fixed point results for nonlinear integral equations

In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and ?iri?’s theorem [1] but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations.     

Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces

In this paper, we extend the coupled fixed point theorems for mixed monotone operators F:X×X→X obtained in [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379–1393] by significantly weakening the contractive condition involved. Our technique of proof is essentially different and more natural. An example as well as an application to periodic BVP is also given in order to illustrate the effectiveness of our generalizations.     

Monotone generalized contractions in partially ordered probabilistic metric spaces

In this paper, a concept of monotone generalized contraction in partially ordered probabilistic metric spaces is introduced and some fixed and common fixed point theorems are proved. Presented theorems extend the results in partially ordered metric spaces of Nieto and Rodriguez-Lopez [Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], Ran and Reurings [A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443] to a more general class of contractive type mappings in partially ordered probabilistic metric spaces and include several recent developments.     

Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces

In this paper, we establish two coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. The theorems presented extend some results due to ?iri? (2009) [3]. An example is given to illustrate the usability of our results.     

Coupled coincidence point theorems in ordered metric spaces

In this paper we extend the coupled contraction mapping theorem proved in partially ordered metric spaces by Gnana Bhaskar and Lakshmikantham (Nonlinear Anal. TMA 65:1379–1393, 2006) to a coupled coincidence point result for a pair of compatible mappings. A control function has been used in our theorem. The mappings are assumed to satisfy a weak contractive inequality. Our theorem improves the results of Harjani et al. (Nonlinear Anal. TMA 74:1749–1760, 2011). The result we have established is illustrated with an example which also shows that the improvement is actual.     

Existence and comparison results for operator and differential equations in abstract spaces

In this paper we apply the chain methods developed in (Heikkilä and Lakshmikantham, 1994) to obtain new fixed point theorems and new existence and comparison results for operator equations in partially ordered sets. These results are then applied to discontinuous implicit functional differential equations in ordered Banach spaces.     

Equivalent conditions for generalized contractions on (ordered) metric spaces

We establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane. As its application, we get that various contractive conditions using the so-called altering distance functions coincide with classical ones. We consider several classes of mappings both on metric spaces and ordered metric spaces. In particular, we show that unexpectedly, some very recent fixed point theorems for generalized contractions on ordered metric spaces obtained by Harjani and Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010) 1188-1197], and Amini-Harandi and Emami [A. Amini-Harandi, H. Emami A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010) 2238-2242] do follow from an earlier result of O’Regan and Petru?el [D. O’Regan and A. Petru?el, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008) 1241-1252].     

Existence results for operator equations in abstract spaces and an application

In this paper we apply the recursion method presented in (Heikkilä and Lakshmikantham, 1994) to derive fixed point theorems and existence results for operator equations in partially ordered sets. The obtained results, combined with related results of (Heikkilä, 2002) are then applied to operator equations in ordered Hilbert spaces and to a functional elliptic boundary value problem.     

模糊度量空间中多变量压缩映射的公共耦合不动点定理

本文提出了多变量映射的一致点的概念.本文的结果是模糊偏序度量空间中不动点定理的一些主要结论从低维到高维的推广.     

Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces

In this paper we introduce the concept of a tripled coincidence point for a pair of nonlinear contractive mappings F : X³ → X and g : X → X. The obtained results extend recent coincidence theorems due to ?iri? and Lakshmikantham [V. Lakshmikantham, L. ?iri?, L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) 4341-4349].     

Suzuki?s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces

Recently, Suzuki [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008) 1861-1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and characterizes the metric completeness. In this paper we prove an analogous fixed point result for a self-mapping on a partial metric space or on a partially ordered metric space. Our results on partially ordered metric spaces generalize and extend some recent results of Ran and Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443], Nieto and Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239]. We deduce, also, common fixed point results for two self-mappings. Moreover, using our results, we obtain a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends Suzuki?s characterization of metric completeness.     

Coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces

In this paper, we introduce the concept of a mixed g-monotone mapping and prove coupled coincidence and coupled common fixed point theorems under ?-contractive conditions for self-maps in partially ordered complete probabilistic metric spaces.     

On fixed point theorems of mixed monotone operators and applications

In this paper, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Finally we apply the main theorem to a nonlinear parabolic partial differential equation. Some results are new even for increasing or decreasing operators.     

Common fixed point theorems under strict contractive conditions in Menger spaces

The main purpose of this paper is to establish some common fixed point theorems under strict contractive conditions for mappings satisfying the property (E.A) in Menger probabilistic metric spaces. As applications, we obtain the corresponding common fixed point theorems under strict contractive in metric spaces.