Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems

In this paper, we first establish the existence theorems of generalized vector quasi-equilibrium problems. From these results, we establish the existence theorems of common fixed point theorems for two multivalued maps and mathematical programs with an equilibrium constraint as applications.     

Variational relation problems and equivalent forms of generalized Fan-Browder fixed point theorem with applications to Stampacchia equilibrium problems

In this paper, we study the existence theorems of solution for variational relation problems. From the existence theorems of solution for variational relation problems, we study equivalent forms of generalized Fan-Browder fixed point theorem, existence theorems of solutions for Stampacchia vector equilibrium problems and generalized Stampacchia vector equilibrium problems. Our results contains many orginal results and have many applications in Nonlinear Analysis.     

Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory

This paper introduces a vectorial form of equilibrium version of Ekeland-type variational principle. Some equivalent results to our variational principle are given. As applications, we derive the existence of solutions of a vector equilibrium problem in the setting of complete quasi-metric spaces with a W-distance. Caristi-Kirk fixed point theorem for multivalued maps is also established in a more general setting.     

A fixed point theorem with applications to truncated lattices

We prove a fixed point theorem related to the set P2 of [17]. The result gives access to nontrivial infinite ordered sets with the fixed point property. We also show how the result can be used to provide an elementary proof of part of Baclawski and Björner’s results on truncated lattices.Dedicated to the memory of Ivan RivalReceived December 1, 2002; accepted in final form June 18, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Iterative methods for generalized equilibrium problems and fixed point problems with applications

In this paper, we consider an iterative method for finding a common element of the set of a generalized equilibrium problem, of the set of solutions to a system of variational inequalities and of the set of fixed points of a strict pseudo-contraction. Strong convergence theorems are established in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.     

Set-valued equilibrium problems with applications to Browder variational inclusions and to fixed point theory

In this paper, we deal with set-valued equilibrium problems under mild conditions of continuity and convexity on subsets recently introduced in the literature. We obtain that neither semicontinuity nor convexity are needed on the whole domain when solving set-valued and single-valued equilibrium problems. As applications, we derive some existence results for Browder variational inclusions, and we extend the well-known Berge maximum theorem in order to obtain two versions of Kakutani and Schauder fixed point theorems.     

Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications

In this paper, we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems, the set of common fixed points of a family of infinite non-expansive mappings. Strong convergence theorems are established in a real Hilbert space under suitable conditions. As some applications, we consider convex feasibility problems and equilibrium problems. The results presented improve and extend the corresponding results of many others.     

Convergence theorems of common elements for equilibrium problems and fixed point problems

In this paper, we introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of two asymptotically nonexpansive mappings in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the two sets.     

与渐近非扩张半群不动点问题相关的分裂等式混合均衡问题及其应用

研究了与渐近非扩张半群不动点问题相关的分裂等式混合均衡问题.在等式约束下,为同时逼近两个空间中混合均衡问题和渐近非扩张半群不动点问题的公共解,借助收缩投影方法引出了一种迭代程序.在适当条件下,该迭代算法的强收敛性被证明.文末还把所得结果应用于分裂等式混合变分不等式问题和分裂等式凸极小化问题.     

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2^Y a point-to-set mapping such that for any χ ε K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ^* ε K such that f g(χ^*), y) -int C(χ) for all y ε K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.     

A new fixed point theorem and its applications

In this paper, we first give a new fixed theorem of lower semicontinuous multivalued mappings, and then, as its applications we obtain some new equilibrium theorems for abstract economies and qualitative games.

     

A NOVEL FIXED POINT THEOREM AND ITS APPLICATIONS

In this article, a novel fixed point theorem in C[0,1] space is established by using the properties of fixed point index. This theorem is then applied to prove the existence of positive solutions for three-point boundary value problems and generalizes some previous results.     

Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory

In this paper, we introduce the concept of a Q$Q$-function defined on a quasi-metric space which generalizes the notion of a τ$\tau$-function and a w$w$-distance. We establish Ekeland-type variational principles in the setting of quasi-metric spaces with a Q$Q$-function. We also present an equilibrium version of the Ekeland-type variational principle in the setting of quasi-metric spaces with a Q$Q$-function. We prove some equivalences of our variational principles with Caristi–Kirk type fixed point theorems for multivalued maps, the Takahashi minimization theorem and some other related results. As applications of our results, we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps. We also extend the Nadler’s fixed point theorem for multivalued maps to a Q$Q$-function and in the setting of complete quasi-metric spaces. As a consequence, we prove the Banach contraction theorem for a Q$Q$-function and in the setting of complete quasi-metric spaces. The results of this paper extend and generalize many results appearing recently in the literature.     

A generalization of Browder’s fixed point theorem with applications

In this paper, the concept of c$c$-compact mapping is introduced. A generalization of Browder’s fixed point theorem and some equivalence forms are given. As applications, the existence of solutions for some variational inequalities and monotone operator equations is discussed.     

Viscosity methods for common solutions for equilibrium and hierarchical fixed point problems

Implicit and explicit viscosity methods for finding common solutions of equilibrium and hierarchical fixed points are presented. These methods are used to solve systems of equilibrium problems and variational inequalities where the involving operators are complements of nonexpansive mappings. The results here are situated on the lines of the research of the corresponding results of Moudafi [Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl. 23 (2007), pp. 1635–1640; Weak convergence theorems for nonexpansive mappings and equilibrium problems, to appear in JNCA], Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl. Art ID 95453 (2006), 10 pp.; Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. Optim. 3 (2007), pp. 529–538; Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, to appear in JNCA], Yao and Liou [Weak and strong convergence of Krasnosel'ski?–Mann iteration for hierarchical fixed point problems, Inverse Probl. 24 (2008), 015015 8 pp.], S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006), pp. 506–515], Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, preprint.], Combettes and Hirstoaga [Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), pp. 117–136] and Plubtieng and Pumbaeang [A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), pp. 455–469.].     

On variational inclusion and common fixed point problems in Hilbert spaces with applications

The purpose of this paper is to introduce a general iterative method for finding a common element of the solution set of quasi-variational inclusion problems and of the common fixed point set of an infinite family of nonexpansive mappings in the framework Hilbert spaces. Strong convergence of the sequences generated by the purposed iterative scheme is obtained.     

Strong convergence theorems for variational inequality, equilibrium and fixed point problems with applications

In this paper, we introduce a new iterative scheme for finding a common element of the set of common solutions of a finite family of equilibrium problems with relaxed monotone mappings, of the set of common solutions of a finite family of variational inequalities and of the set of common fixed points of an infinite family of nonexpansive mappings in a Hilbert space. Strong convergence for the proposed iterative scheme is proved. As an application, we solve a multi-objective optimization problem using the result of this paper. Our results improve and extend the corresponding ones announced by others.     

A fixed point theorem

We establish a fixed point theorem for a Lie group of isometries acting on a Riemannian manifold with nonnegative curvature.     

On split common fixed point problems

Based on the convergence theorem recently proved by the second author, we modify the iterative scheme studied by Moudafi for quasi-nonexpansive operators to obtain strong convergence to a solution of the split common fixed point problem. It is noted that Moudafi's original scheme can conclude only weak convergence. As a consequence, we obtain strong convergence theorems for split variational inequality problems for Lipschitz continuous and monotone operators, split common null point problems for maximal monotone operators, and Moudafi's split feasibility problem.