A Penalty Approach for Generalized Nash Equilibrium Problem

The generalized Nash equilibrium problem （GNEP） is a generalization of the standard Nash equilibrium problem （NEP）, in which both the utility function and the strategy space of each player depend on the strategies chosen by all other players. This problem has been used to model various problems in applications. However, the convergent solution algorithms are extremely scare in the literature. In this paper, we present an incremental penalty method for the GNEP, and show that a solution of the GNEP can be found by solving a sequence of smooth NEPs. We then apply the semismooth Newton method with Armijo line search to solve latter problems and provide some results of numerical experiments to illustrate the proposed approach.

Behavior of solution set for bilevel generalized mixed equilibrium problems in topological vector spaces

A new bilevel generalized mixed equilibrium problem （BGMEF） is introduced and studied in topological vector spaces. By using a minimax inequality, the existence of solutions and the behavior of solution set for the BGMEP are studied under quite mild conditions. These results are new and generalize some recent results in this field.

SOME RESULTS ON GENERALIZED EQUILIBRIUM PROBLEMS INVOLVING STRICTLY PSEUDOCONTRACTIVE MAPPINGS

In this paper,we consider an iterative sequence for generalized equilibrium problems and strictly pseudocontractive mappings.We show that the iterative sequence converges strongly to a common element of the solution set of generalized equilibrium problems and of the fixed point set of strictly pseudocontractive mappings.

Generalized mixed equilibrium problem in Banach spaces

This paper uses a hybrid algorithm to find a common element of the set of solutions to a generalized mixed equilibrium problem, the set of solutions to variational inequality problems, and the set of common fixed points for a finite family of quasi-C- nonexpansive mappings in a uniformly smooth and strictly convex Banach space. As applications, we utilize our results to study the optimization problem. It shows that our results improve and extend the corresponding results announced by many others recently.

A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM,FIXED POINT PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS,AND QUASI-VARIATIONAL INCLUSION PROBLEM

In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseudo-contractive mappings, and the set of solutions of the variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extends the recent results in G.L.Acedo and H.K.Xu [2], Zhang, Lee and Chan [8], Wakahashi and Toyoda [9], Takahashi and Takahashi [I0] and S. S. Chang, H. W. Joseph Lee and C. K. Chan [II], S.Takahashi and W.Takahashi [12]. Moreover, the method of proof adopted in this article is different from those of [4] and [12].

EQUILIBRIUM PROBLEMS WITH LOWER AND UPPER BOUNDS IN TOPOLOGICAL SPACES

By employing a fixed point theorem due to Ding, Park and Jung, some existence theorems of solutions for equilibrium problems with lower and upper bounds are proved in noncompact topological spaces. These results further answer the open problem raised by Isac, Sehgal and Singh under much weaker assumptions.

NONEMPTY INTERSECTION THEOREMS AND SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS IN PRODUCT G-CONVEX SPACES

By using an existence theorems of maximal elements for a family of set-valued mappings in G-convex spaces due to the author, some new nonempty intersection theorems for a family of set-valued mappings were established in noncompact product G-convex spaces. As applications, some equilibrium existence theorems for a system of generalized vector equilibrium problems were proved in noncompact product G-convex spaces. These theorems unify, improve and generalize some important known results in literature.

EKELAND'S PRINCIPLE FOR SET-VALUED VECTOR EQUILIBRIUM PROBLEMS

In this paper, we introduce a concept of quasi C-lower semicontinuity for setvalued mapping and provide a vector version of Ekeland＇s theorem related to set-valued vector equilibrium problems. As applications, we derive an existence theorem of weakly efficient solution for set-valued vector equilibrium problems without the assumption of convexity of the constraint set and the assumptions of convexity and monotonicity of the set-valued mapping. We also obtain an existence theorem of ε-approximate solution for set-valued vector equilibrium problems without the assumptions of compactness and convexity of the constraint set.

WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES

In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.

A NEW CLASS OF BILEVEL GENERALIZED MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES

A new class of bilcvel generalized mixed equilibrium problems involving setvalued mappings is introduced and studied in a real Banach space. By using the auxiliary principle technique, new iterative algorithms for solving the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems involving set-valued mappings are suggested and analyzed. Existence of solutions and strong convergence of the iterative sequences generated by the algorithms are proved under quite mild conditions. The behavior of the solution set of the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems is also discussed. These results are new and generalize some recent results in this field.

LEVITIN-POYAK WELL-POSEDNESS OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS WITH FUNCTIONAL CONSTRAINTS

In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.

SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS ON PRODUCT FC-SPACES

By applying a maximal element theorem on product FC-space due to author, some new equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in FC-spaces. By using these equilibrium existence theorems, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are established in noncompact product FC-spaces. These results improve and generalize some recent results in literature to product FC-spaces without any convexity structure.

QUASI-EQUILIBRIUM PROBLEMS AND CONSTRAINED MULTIOBJECTIVE GAMES IN GENERALIZED CONVEX SPACE

A class of quasi-equilibrium problems and a class of constrained multiobjective games were introduced and studied in generalized convex spaces without linear structure. First, two existence theorems of solutions for quasi-equilibrium problems are proved in noncompact generalized convex spaces. Then, ar applications of the quasi-equilibrium existence theorem, several existence theorems of weighted Nash-equilibria and Pareto equilibria for the constrained multiobjective games are established in noncompact generalized convex spaces. These theorems improve, unify, and generalize the corresponding results of the multiobjective games in recent literatures.

A SOLUTION OF A GENERAL EQUILIBRIUM PROBLEM

Under the framework of a real Hilbert space, we introduce a new iterative method for finding a common element of the set of solution of a general equilibrium problem and the set of fixed points of a nonexpansive semigroup. Moreover, a numerical example is presented. This example grantee the main result of the paper.

EXISTENCE RESULTS FOR GLOBALLY EFFICIENT SOLUTIONS OF VECTOR EQUILIBRIUM PROBLEMS VIA A GENERALIZED KKM PRINCIPLE

The aim of this article is to present new existence results for globally efficient solutions of a strong vector equilibrium problem given by a sum of two functions via a generalized KKM principle, and to establish the connectedness of the solutions set.

QUASI_EQUILIBRIUM PROBLEMS IN NONCOMPACT GENERALIZED CONVEX SPACES

By applying a new fixed point theorem due to the author, some new equilibrium existence theorems of quasi-equilibrium problems are proved in noncompact generalized convex spaces. These theorems improve and generalize a number of important known results in recent literature.

NONEMPTY INTERSECTION THEOREMS AND SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS IN PRODUCT G-CONVEX SPACES

By using an existence theorems of maximal elements for a family of set-valued mappings in G-convex spaces due to the author, some new nonempty intersection theorems for a family of set-valued mappings were established in noncompact product G-convex spaces. As applications, some equilibrium existence theorems for a system of generalized vector equilibrium problems were proved in noncompact product G-convex spaces. These theorems unify, improve and generalize some important known results in literature.

GENERIC STABILITY AND EXISTENCE OF ESSENTIAL COMPONENTS OF THE SOLUTION SET FOR THE SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS

By using Fort theorem the generic stability result for the system of generalized vector equilibrium problems is established. Further, by proving the existence and connectivity of minimal essential set the existence result of essential components in the solution set is derived.

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.

Strong convergence theorem for a generalized equilibrium problem and a k-strict pseudocontraction in Hilbert spaces

The main purpose of this paper is to study a new iterative algorithm for finding a common element of the set of solutions for a generalized equilibrium problem and the set of fixed points for a k-strict pseudocontractive mapping in the Hilbert space. The presented results extend and improve the corresponding results reported in the lit-erature.