Generalized 2-KKM theorems and their applications in hyperconvex metric spaces

In this work, we first define the 2-KKM mapping and the generalized 2-KKM mapping on a metric space, and then we apply the property of the hyperconvex metric space to get a KKM theorem and a fixed point theorem without a compactness assumption. Next, by using this KKM theorem, we get some variational inequality theorems and minimax inequality theorems.     

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G$G$-convex spaces properly. In this work, we show that a large number of results in KKM theory on G$G$-convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.     

Generalized selection theorems without convexity

In this paper, we extend new selection theorems for almost lower semicontinuous multifunctions T on a paracompact topological space X to general nonconvex settings. On the basis of the Kim-Lee theorem and the Horvath selection theorem, we first show that any a.l.s.c. C-valued multifunction admits a continuous selection under a mild condition of a one-point extension property. Finally, we apply a fundamental selection theorem, due to Ben-El-Mechaiekh and Oudadess, to modify our selection theorems by adjusting a closed subset Z of X with its covering dimension dim_XZ≤0. The results derived here generalize and unify various earlier ones from classic continuous selection theory.     

The KKM principle in abstract convex spaces: Equivalent formulations and applications

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its “open” version. In this paper, we clearly derive a sequence of a dozen statements which characterize the KKM spaces and equivalent formulations of the partial KKM principle. As their applications, we add more than a dozen statements including generalized formulations of von Neumann minimax theorem, von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.     

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.     

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.     

Endpoints of set-valued contractions in metric spaces

Suppose (X,d) be a complete metric space, and suppose F:X→CB(X) be a set-valued map satisfies H(Fx,Fy)≤ψ(d(x,y)), , where ψ:[0,∞)→[0,∞) is upper semicontinuous, ψ(t)<t for each t>0 and satisfies lim inf_t→∞(t−ψ(t))>0. Then F has a unique endpoint if and only if F has the approximate endpoint property.     

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.     

Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces

In hyperconvex metric spaces we consider best approximation, invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive.     

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.     

Multi-valued contraction mappings in metric spaces

Summary Some fixed point theorems for multi-valued contraction mappings in metric spaces, related to a fixed point theorem due to T. Zamfirescu [6], [5], are presented.     

Relative Nielsen theory for noncompact spaces and maps

In this note, we generalize the various existing local and relative Nielsen type numbers to the setting of maps of noncompact ANR-pairs. Then we introduce general classes of admissible maps for which these numbers are well-defined. An application of these relative Nielsen numbers to differential equations is also given.     

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.     

On cone metric spaces: A survey

Using an old M. Krein’s result and a result concerning symmetric spaces from [S. Radenovi?, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (1) (2011), 38-50], we show in a very short way that all fixed point results in cone metric spaces obtained recently, in which the assumption that the underlying cone is normal and solid is present, can be reduced to the corresponding results in metric spaces. On the other hand, when we deal with non-normal solid cones, this is not possible. In the recent paper [M.A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl. 2010, 7 pages, Article ID 315398, doi:10.1115/2010/315398] the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant; also that underlying Banach space and the associated cone subset are not necessary. In fact, Khamsi’s approach includes a small class of results and is very limited since it requires only normal cones, so that all results with non-normal cones (which are proper extensions of the corresponding results for metric spaces) cannot be dealt with by his approach.     

Maximal elements in topological and generalized metric spaces

In this note, some problems concerning existence of maximal elements in a topological as well as in a generalized metric space, equipped with an ordering, are studied. The results presented here may be considered as a partial refinement of those established in [2] for uniform structures.     

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.     

The constrained degree and fixed-point index theory for set-valued maps

In the first part of the paper we give a construction of a topological degree theory for set-valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a Banach space. The obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds. In the second part we propose a fixed-point index for inward maps on arbitrary closed convex sets.     

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.     

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.     

New generalizations of basic theorems in the KKM theory

In the present paper, we obtain a new KKM type theorem for intersectionally closed-valued KKM maps and some useful new basic consequences. Typical examples of them are abstract forms of Fan’s matching theorem, Fan’s geometric lemma, the Fan-Browder fixed point theorem, maximal element theorems, Fan’s minimax inequality, variational inequalities, and others.