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.     

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.     

Coupled fixed points in partially ordered metric spaces and application

In this paper, we prove some coupled fixed point theorems for mappings having a mixed monotone property in partially ordered metric spaces. The main results of this paper are generalizations of the main results 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]. As an application, we discuss the existence and uniqueness for a solution of a nonlinear integral equation.     

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.     

Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations

In this paper, coincidence and common fixed point results are established in a partially ordered cone metric space. An application of our results obtained to prove the existence of a common solution to integral equations is presented.     

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].     

Fixed point theorems for generalized contractions on partial metric spaces

We obtain two fixed point theorems for complete partial metric space that, by one hand, clarify and improve some results that have been recently published in Topology and its Applications, and, on the other hand, generalize in several directions the celebrated Boyd and Wong fixed point theorem and Matkowski fixed point theorem, respectively.     

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.     

Note on “Coupled fixed point theorems for contractions in fuzzy metric spaces”

In the paper “Coupled fixed point theorems for contractions in fuzzy metric spaces” by Sedghi et al. [S. Sedghi, I. Altun, N. Shobec, Coupled fixed point theorems for contractions in fuzzy metric spaces, Nonlinear Analysis 72 (2010) 1298-1304], a coupled common fixed point result was presented. However, our purpose is to show that this result and its proof are false. We give a counterexample and also explain how to correct this result. As a modification, we state and prove a coupled fixed point theorem under some hypotheses of fuzzy metric and t-norm.     

Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces

Generalizations of the Edelstein-Suzuki theorem [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal. TMA 71 (2009), 5313-5317], including versions of the Kannan, Chatterjea and Hardy-Rogers-type fixed point results for compact metric spaces, are proved. Also, abstract metric versions of these results are obtained. Examples are presented to distinguish our results from the existing ones.     

Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric

In this paper, we study the existence and uniqueness of (coupled) fixed points for mixed monotone mappings in partially ordered metric spaces with semi-monotone metric. As an application, we prove the existence and uniqueness of the solution for a first-order differential equation with periodic boundary conditions.     

On coincidence point and fixed point theorems for nonlinear multivalued maps

Several characterizations of MT-functions are first given in this paper. Applying the characterizations of MT-functions, we establish some existence theorems for coincidence point and fixed point in complete metric spaces. From these results, we can obtain new generalizations of Berinde-Berinde?s fixed point theorem and Mizoguchi-Takahashi?s fixed point theorem for nonlinear multivalued contractive maps. Our results generalize and improve some main results in the literature.     

Fixed point results for generalized quasicontraction mappings in abstract metric spaces

In this paper, we introduce the concept of generalized quasicontraction mappings in an abstract metric space. By using this concept, we construct an iterative process which converges to a unique fixed point of these mappings. The result presented in this paper generalizes the Banach contraction principle in the setting of metric space and a recent result of Huang-Zhang for contractions. We also validate our main result by an example.     

Fixed point theorems for generalized weakly contractive mappings

We prove a fixed point theorem for a generalized weakly contractive mapping and a fixed point theorem for a pair of weakly contractive mappings. We also show that these mappings satisfy properties P and Q.     

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.     

Fixed point results for set-valued contractions by altering distances in complete metric spaces

Nadler’s contraction principle has led to fixed point theory of set-valued contraction in non-linear analysis. Inspired by the results of Nadler, the fixed point theory of set-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi–Takahashi, Feng–Liu and many others. In the present paper, the concept of generalized contractions for set-valued maps in metric spaces is introduced and the existence of fixed point for such a contraction are guaranteed by certain conditions. Our first result extends and generalizes the Nadler, Feng–Liu and Klim–Wardowski theorems and the second result is different from the Reich and Mizoguchi–Takahashi results. As a consequence, we derive some results related to fixed point of set-valued maps satisfying certain conditions of integral type.     

Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions

In this paper, we obtain some common fixed point theorems for occasionally weakly compatible mappings on a set X together with the function d:X×X→[0,∞) without using the triangle inequality and assuming symmetry only on the set of points of coincidence.     

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].     

Ekeland's variational principle and Caristi's coincidence theorem for set-valued mappings in probabilistic metric spaces

By using the partial ordering method, a more general type of Ekeland's variational principle and Caristi's coincidence theorem for set-valued mappings in probabilistic metric spaces are given in this paper. In addition, we give also a directly simple proof of the equivalence between theses theorems in probabilistic metric spaces.This paper was supported financially from the National Natural Science Foundation of China, 1994, and the Basic Science Research Institute Program, Ministry of Education, Korea, 1995, Project No. BSRI-95-1405.     

Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces

Let (X,?) be a partially ordered set and d be a complete metric on X. Let F,G be two set-valued mappings on X. We obtained sufficient conditions for the existence of common fixed point of F and G satisfying an implicit relation in partially ordered set X.