Viscosity approximation methods with weakly contractive mappings for nonexpansive mappings

Let K be a closed convex subset of a real reflexive Banach space E, T：K→K be a nonexpansive mapping, and f：K→K be a fixed weakly contractive （may not be contractive） mapping. Then for any t∈（0, 1）, let x1∈K be the unique fixed point of the weak contraction x1→tf（x）＋（1-t）Tx. If T has a fixed point and E admits a weakly sequentially continuous duality mapping from E to E^＊, then it is shown that {xt} converges to a fixed point of T as t→0. The results presented here improve and generalize the corresponding results in （Xu, 2004）.

Weakly R-KKM mappings-intersection theorems and minimax inequalities in topological spaces

In this paper, we introduce the concepts of weakly R-KKM mappings, R- convex and R-β-quasiconvex in general topological spaces without any convex structure. Relating to these, we obtain an extension to general topological spaces of Fan＇s matching theorem, namely that Lemma 1.2 in this paper. On this basis, two intersection theorems are proved in topological spaces. By using intersection theorems, some minimax inequalities of Ky Fan type are also proved in topological spaces. Our results generalize and improve the corresponding results in the literature.

FIXED POINT INDEX THEORY FOR WEAKLY INWARD MAPPINGS

In this paper we define a fixed point index theory for locally Lip., completely continuous and weakly inward mappings defined on closed convex sets in general Banach spaces where no other artificial conditions are imposed. This makes ns to deal with these kinds of mappings more easily. As obvious applications, some results in [3],[5],[7],[9],[10] are deepened and extended.

FIXED POINTS FOR WEAKLY INWARD MAPPINGS IN BANACH SPACES (Ⅱ)

S. Hu and Y. Sun[1] defined the fixed point index for weakly inward mappings, investigated its properties and studied fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we further investigate boundary conditions, under which the fixed point index for i（A, Ω, p） is equal to nonzero, where i（A, Ω, p） is the completely continuous and weakly inward mapping. Correspondingly, we can obtain many new fixed point theorems of the completely continuous and weakly inward mapping, which generalize some famous theorems such as Rothe＇s theorem, Altman＇s theorem, Petryshyn＇s theorem etc. in the case of weakly inward mappings. In addition, our conclusions extend the famous fixed point theorem of cone expansion and compression to the case of weakly inward mappings. Moreover, the main results contain and generalize the corresponding results in the recent work[2].

A Common Fixed Point Theorem for Weakly Compatible Mappingsin Fuzzy Semi-Metric Space

In this paper, we establish a common fixed pointtheorem for three pairs of self-mappings in fuzzy semi-metric space which improves and extends similar known results in the literature.

ODEs IN BANACH SPACES AND FIXED POINTS OF WEAKLY INWARD MAPS

This paper is an attempt to investigate systematically fixed points of weakly inward maps by using some basic results from differential equations in Banach spaces. By investigating the Poincare operators for such differential equations, we establish a fixed point index theory for two classes of weakly inward maps.

Some Common Fixed Point Theorems for Four Mappings in Dislocated Metric Space

In this article, we establish some common fixed point theorems for two pairs of weakly compatible mappings with (E. A.) and (CLR) property in dislocated metric space which generalize and extend some similar results in the literature.

Viscosity approximation with weak contractions for fixed point problem,equilibrium problem,and variational inequality problem

This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction.The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality.Under suitable conditions,some strong convergence theorems are established in the framework of Hilbert spaces.The results presented in the paper improve and extend the corresponding results of Colao et al.（Colao,V.,Acedo,G.L.,and Marino,G.An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings.Nonlinear Anal.71,2708–2715（2009））,Plubtieng and Punpaeng（Plubtieng,S.and Punpaeng,R.A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.J.Math.Anal.Appl.336,455–469（2007））,Colao et al.（Colao,V.,Marino,G.,and Xu,H.K.An iterative method for finding common solutions of equilibrium problem and fixed point problems.J.Math.Anal.Appl.344,340–352（2008））,Yao et al.（Yao,Y.,Liou,Y.C.,and Yao,J.C.Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings.Fixed Point Theory Application 2007,Article ID 64363（2007）DOI 10.1155/2007/64363）,and others.

ON PROBLEM OF NEAREST COMMON FIXED POINT OF NONEXPANSIVE MAPPINGS

The purpose is by using the viscosity approximation method to study the convergence problem of the iterative scheme for an infinite family of nonexpansive mappings and a given contractive mapping in a reflexive Banach space. Under suitable conditions, it was proved that the iterative sequence converges strongly to a common fixed point which was also the unique solution of some variational inequality in a reflexive Banach space. The results presented extend and improve some recent results.

Weak KKM theorems in generalized finitely continuous space with applications

Some new weak Knaster-Kuratouski-Mazurkiewicz （KKM） theorems are proved under the noncompact situation in the generalized finitely continuous space （GFC-space） without any convexity. As applications, the minimax inequalities of the Ky Fan type are also given under some suitable conditions. The results unify and generalize some known results in recent literatures.

Weak Graph Map Homotopy and Its Applications

The authors introduce a notion of a weak graph map homotopy(they call it M-homotopy),discuss its properties and applications.They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001.The difference between them is that the weak graph map homotopy transformation is defined in terms of maps,while the graph homotopy transformation is defined by means of combinatorial operations.They discuss its advantages over the graph homotopy transformation.As its applications,they investigate the mapping class group of a graph and the 1-order M P-homotopy group of a pointed simple graph.Moreover,they show that the 1-order M P-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.

Strong convergence theorems for nonexpansive semi-groups in Banach spaces

Some strong convergence theorems of explicit composite iteration scheme for nonexpansive semi-groups in the framework of Banach spaces are established. Results presented in the paper not only extend and improve the corresponding results of ShiojiTakahashi, Suzuki, Xu and Aleyner-Reich, but also give a partially affirmative answer to the open questions raised by Suzuki and Xu.

Generalized KKM-Type Theorems for Weakly Generalized KKM Mapping with Some Applications

In this paper, we establish some new generalized KKM-type theorems based on weakly generalized KKM mapping without any convexity structure in topological spaces. As applications, some minimax inequalities and an existence theorem of equilibrium points for an abstract generalized vector equilibrium problem are proved in topological spaces. The results presented in this paper unify and generalize some known results in recent literature.

Points of Coincidence for Weakly Compatible Pair of Mappings in Cone Metric Spaces

In this paper, some new existence and uniqueness of points of coincidence of weakly compatible pair of mappings is obtained, which does not satisfy continuity and commutativity. The conditions are weaker than the usual conditions in cone metric spaces.

Some Integral Type Fixed Point Theorems in Dislocated Metric Space

In this article, we establish a common fixed point theorem satisfying integral type contractive condition for two pairs of weakly compatible mappings with E. A. property and also generalize Theorem (2) of B.E. Rhoades [1] in dislocated metric space.

Regularity for weakly(K_(1),K_(2))-quasiregular mappings

In this paper, we first give the definition of weakly (K1,K2-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse H?lder inequality, we obtain their regularity property: For anyq 1 that satisfies $0< K_1 n^{(n + 4)/2} 2^{n + 1} \times 100^{n^2 } [2^{3n/2} (2^{5n} + 1)](n - q_1 )< 1$ , there existsp 1=p 1(n,q 1,K 1,K 2)>n, such that any (K1, K2)-quasiregular mapping $f \in W_{loc}^{1,q_1 } (\Omega ,R^n )$ is in fact in $W_{loc}^{1,p_1 } (\Omega , R^n )$ . That is, f is (K1,K2)-quasiregular in the usual sense.

Hybrid projection method for generalized mixed equilibrium problems,variational inequality problems,and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.

Existence of Invariant Tori for Certain Weakly Reversible Mappings

In this paper, we consider certain mappings. M, sufficiently close to an integrable one, which is weakly reversible with respect to the mappings G sufficiently close to an involution of type (m.n). where m, n∈Z_+ are arbitrary. Under some weak non-degeneracy condition, we construct a uniform KAM iteration for proving the existence of a Cantor family of m-tori invariant under the reversible mappings M and the reversing mapping G.

Fixed Point Theorems for Weakly Contractive Mappings in Ordered Metric Spaces with an Application

In this paper,we prove fixed point theorem for weakly contractive mappings using locally T-transitivity of binary relation and presenting an analogous version of Harjani and Sadarangani theorem involving more general relation theoretic metrical notions.Our fixed point results under universal relation reduces to Harjani and Sadarangani[Nonlinear Anal.,71(2009),3403-3410]fixed point theorems.In this way we also generalize some of the recent fixed point theorems for weak contraction in the existing literature.

Coincidence Points and Almost-like Coincidence Points on Generalized ConvexSpaces with Applications

We introduce the concept of a weakly G-quasiconvex map with respect to a map on generalized convex spaces and use the concept to prove coincidence point theorems and almost-like coincidence point theorems. As applications of the above results, we derive almost fixed point theorems and fixed point theorem. These main results generalize and improve some known results in the literature.