Some results on generalized equilibrium problems involving a family of nonexpansive mappings

In this paper, we introduce an iterative method for finding a common element in the solution set of generalized equilibrium problems, in the solution set of variational inequalities and in the common fixed point set of a family of nonexpansive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.     

Gwinner变分不等式和隐变分不等式

本文在研究Gwinner变分不等式的基础上，利用新的集值映象不动点定理，探讨一类具广泛意义的隐变分不等式问题，改进了迄今相关结果中对紧性条件的要求，在非紧设置下获得了解的存在性定理．     

拓扑型截口定理及应用*

本文给出一个新型的KKM定理,并用它得到拓扑型截口定理,在第四节至第五节应用此截口定理给出了Browder-Hartman-Stampacchia变分不等式[3].隐变分不等式[8],抽象形式变分不等式[19]的解的存在性定理,和一个集值映射的不动点定理。其结果不仅包含TBrowder[3]中的主要结果为特例,而且,改进和发展了引文[1～19]中的相应结果。     

A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings

In this paper, we investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for k-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of Kumam [P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turk. J. Math. 33 (2009) 85–98], Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications, 2008, Article ID 134148, 17 pages, doi:10.1155/2008/134148], Yao et al. [Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an finite family of nonexpansive mappings, Nonlinear Analysis 69 (5–6) (2008) 1644–1654], Qin et al. [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis (69) (2008) 3897–3909], and many others.     

多值单调映象的Browder-Hartman-Stampacchia型变分不等式

在Ｂａｎａｃｈ空间中,证明了多值单调映象的Ｂｒｏｗｄｅｒ－Ｈａｒｔｍａｎ－Ｓｔａｍｐａｃｃｈｉａ型变分不等式解的存在性定理,统一并推广了「４,５,６,９」中的主要结果。     

Dynamic thermoviscoelastic problem with friction and damage

In this paper we present a model of dynamic frictional contact between a thermoviscoelastic body and a foundation. The thermoviscoelastic constitutive law includes a temperature effect described by the parabolic equation with the subdifferential boundary condition and a damage effect described by the parabolic inclusion with the homogeneous Neumann boundary condition. Contact is modeled with bilateral condition and is associated to a subdifferential frictional law. The variational formulation of the problem leads to a system of hyperbolic hemivariational inequality for the displacement, parabolic hemivariational inequality for the temperature and parabolic variational inequality for the damage. The existence of a unique weak solution is proved by using recent results from the theory of hemivariational inequalities, variational inequalities, and a fixed point argument.     

Approximation of common solutions of variational inequalities via strict pseudocontractions

In this paper, a convex feasibility problem is considered. We construct an iterative method to approximate a common element of the solution set of classical variational inequalities and of the fixed point set of a strict pseudocontraction. Strong convergence theorems for the common element are established in the framework of Hilbert spaces.     

Nonzero solutions for generalized variational inequalities by index

The existence of nonzero solutions for a class of variational inequalities is studied by fixed point index approach for multivalued mappings in finite dimensional spaces and reflexive Banach spaces.     

Equivalent forms of a generalized KKM theorem and their applications

In this paper, we study various types of variational relation problems. We establish the existence of solutions for these types of problems and point out some important particular cases and their applications. We also show that some existence theorems of solution for these types of problems and some existence theorems of variational inclusion problems are equivalent to a generalized KKM theorem. Applying our results we obtain existence theorems of common fixed point, generalized maximal element theorems, a generalized coincidence theorems and a section theorem.     

Existence of nonzero solutions for a class of generalized variational inequalities

The existence of nonzero solutions for a class of generalized variational inequalities is studied by topological degree theory for multi-valued mappings in finite dimensional spaces and reflexive Banach space. One of the mappings concerned here is nonlinear with coercive or monotone and other is set-contractive or upper semi-continuous. Under some suitable assumptions, some existence theorems of nonzero solutions for this generalized variational inequalities are obtained. This work was supported by the Young Talent Foundation of Zhejiang Gongshang University and the Foundation of Department of Education of Zhejiang Province No. 20070628.     

一类积集上的广义向量变分不等式

研究一类积集上具某种权向量的广义向量变分不等式组及其广义向量变分不等式的有关问题,刻画它们之间解的相互关系.在映射的次连续性和关于某向量广义单调性的条件下,利用集值映射的不动点定理,对所讨论的几种类型的广义向量变分不等式给出解的存在性.     

Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications

In this paper, we consider the problem of convergence of an iterative algorithm for a system of generalized variational inequalities and a nonexpansive mapping. Strong convergence theorems are established in the framework of real Banach spaces.     

Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications

We use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi-Lipschitzian mappings. As a consequence, several convergence theorems for quasi-nonexpansive mappings and asymptotically κ-strict pseudo-contractions are deduced. We also establish strong convergence of iterative sequences for finding a common element of the set of fixed point, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain the corresponding results due to Tada-Takahashi and Nakajo-Shimoji-Takahashi.     

Minimization principles for elliptic hemivariational inequalities

In this paper, we explore conditions under which certain elliptic hemivariational inequalities permit equivalent minimization principles. It is shown that for an elliptic variational–hemivariational inequality, under the usual assumptions that guarantee the solution existence and uniqueness, if an additional condition is satisfied, the solution of the variational–hemivariational inequality is also the minimizer of a corresponding energy functional. Then, two variants of the equivalence result are given, that are more convenient to use for applications in contact mechanics and in numerical analysis of the variational–hemivariational inequality. When the convex terms are dropped, the results on the elliptic variational–hemivariational inequalities are reduced to that on “pure” elliptic hemivariational inequalities. Finally, two representative examples from contact mechanics are discussed to illustrate application of the theoretical results.     

关于广义变分不等式与广义拟变分不等式

本文在非常一般的框架下,建立了极大极小不等式,广义变分不等式和广义拟变分不等式,证明了解的存在定理,且它们是在非紧集上得到的,从而推广和改进了[3～13]中的相应结果.     

Weak convergence theorems for a countable family of Lipschitzian mappings

This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37–43], Tada–Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 (2007) 359–370], and Plubtieng–Kumam [S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings. J. Comput. Appl. Math. (2008) doi:10.1016/j.cam.2008.05.045]. Some of our results are established with weaker assumptions.     

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 vector variational inequalities with star-pseudomonotone and discontinuous operators

It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed.     

Algorithms for Solutions of Variational Inequalities in the Set of Common Fixed Points of Finite Family of λ-Strictly Pseudocontractive Mappings

In this article, we introduce an algorithm which has strong convergence for solving the variational inequality problem for η-inverse strongly accretive mappings in the set of common fixed points of finite family of λ-strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.     

General algorithm of random solutions for random nonlinear variational inequalities in banach Spaces *

In this paper, we study a class of random nonlinear variational inequalities in Banach spaces. By applying a random minimax inequahty obtained by Tarafdar and Yuan, some existence uniqueness theorems of random solutions for the random nonhnear variational inequalities are proved. Next, by applying the random auxiliary problem technique, we suggest an innovative iterative algorithm to compute the random approximate solutions of the random nonlinear variational inequahty. Finally, the convergence criteria is also discussed