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.

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.

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.

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.

ALGORITHMS FOR A SYSTEM OF GENERALIZED MIXED EQUILIBRIUM PROBLEMS AND A COUNTABLE FAMILY OF SOME NONLINEAR MULTI-VALUED NONEXPANSIVE-TYPE MAPS

In this paper, relaxed iterative algorithms of Krasnoselskii-type and Halpern-type that approximate a solution of a system of a generalized mixed equilibrium problem anda common fixed point of a countable family of totally quasi-C-asymptotically nonexpansivemulti-valued maps are constructed. Strong convergence of the sequence generated by thesealgorithms is proved in uniformly smooth and strictly convex real Banach spaces with Kadec-Klee property. Furthermore, several applications of our theorems are also presented. Finally,our theorems are significant improvements on several important recent results for this classof nonlinear problems.

EXISTENCE OF STRONGLY VALID TOLLS FOR MULTICLASS NETWORK EQUILIBRIUM PROBLEMS

In this article, we consider the multiclass network equilibrium problems. A so called strongly valid toll can support any multiclass user equilibrium flow pattern as a system minimum when the system objective function is measured by total emission. Using Hoffman lemma and exact penalization method, we provide the existence of strongly valid tolls for multiclass network equilibrium problems.

Subgradient Extragradient Methods for Equilibrium Problems and Fixed Point Problems in Hilbert Space

Inspired by inertial methods and extragradient algorithms,two algorithms were proposed to investigate fixed point problem of quasinonexpansive mapping and pseudomonotone equilibrium problem in this study.In order to enhance the speed of the convergence and reduce computational cost,the algorithms used a new step size and a cutting hyperplane.The first algorithm was proved to be weak convergence,while the second algorithm used a modified version of Halpern iteration to obtain strong convergence.Finally,numerical experiments on several specific problems and comparisons with other algorithms verified the superiority of the proposed algorithms.

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.

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.

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.

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.

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.

CONVERGENCE ANALYSIS FOR SYSTEM OF EQUILIBRIUM PROBLEMS AND LEFT BREGMAN STRONGLY RELATIVELY NONEXPANSIVE MAPPING

In this paper, we prove strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces. We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.

Bilevel generalized mixed equilibrium problems involving generalized mixed variational-like inequality problems in reflexive Banach spaces

A new bilevel generalized mixed equilibrium problem （BGMEP） involving generalized mixed variational-like inequality problems （GMVLIPs） is introduced and studied in the reflexive Banach spaces. First, an auxiliary generalized mixed equilibrium problem （AGMEP） is introduced to compute the approximate solutions of the BGMEP involving the GMVLIPs. By using a minimax inequality, the existence and the unique- ness of solutions of the AGMEP are proved under mild conditions without any coercive assumptions. By using an auxiliary principle technique, the new iterative algorithms are proposed and analyzed, with which the approximate solutions of the BGMEP are computed. The strong convergence of the iterative sequence generated by the algorithms is shown under mild conditions without any coercive assumptions. These new results can generalize some recent results in this field.

Existence and algorithm of solutions for a system of generalized mixed implicit equilibrium problems in Banach spaces

A new system of generalized mixed implicit equilibrium problems is introduced and studied in Banach spaces. First, the notion of the Yosida proximal mapping for generalized mixed implicit equilibrium problems is introduced. By using the notion, a system of generalized equation problems is considered, and its equivalence with the system of generalized mixed implicit equilibrium problems is also proved. Next, by applying the system of generalized equation problems, we suggest and analyze an iterative algorithm to compute the approximate solutions of the system of generalized mixed implicit equilibrium problems. The strong convergence of the iterative sequences generated by the algorithm is proved under quite mild conditions. The results are new and unify and generalize some recent results in this field.

Existence results for generalized vector equilibrium problems with applications

By a coincidence theorem, some existence theorems of solutions are proved for four types of generalized vector equilibrium problems with moving cones. Applications to the generalized semi-infinite programs with the generalized vector equilibrium constraints under the mild conditions are also given. The results of this paper unify and improve the corresponding results in the previous literature.

Auxiliary principle and approximation solvability for system of new generalized mixed equilibrium problems in reflexive Banach spaces

A new system of generalized mixed equilibrium problems （SGMEPs） involving generalized mixed variational-like inequality problems is introduced and studied in reflexive Banach spaces. A system of auxiliary generalized mixed equilibrium problems （SAGMEPs） for solving the SGMEPs is first introduced. Existence and uniqueness of the solutions to the SAGMEPs is proved under quite mild assumptions without any coercive conditions in reflexive Banach spaces. Using the auxiliary principle technique, a new iterative algorithm for solving the SGMEPs is proposed and analyzed. Strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild assumptions without any coercive conditions. These results improve, unify, and generalize some recent results in this field.

Continuity of the Solution Mappings for Parametric Generalized Strong Vector Equilibrium Problems

The stability analysis of the solution mappings for vector equilibrium problems is an important topic in optimization theory and its applications. In this paper, we focus on the continuity of the solution mapping for a parametric generalized strong vector equilibrium problem. By virtue of a nonlinear scalarization technique, a new density result of the solution mapping is obtained. Based on the density result, we give sufficient conditions for the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mapping to the parametric generalized strong vector equilibrium problem. In addition, some examples were given to illustrate that our results improve ones in the literature.

Vectorial Ekeland variational principle for systems of equilibrium problems and its applications

For a family of vector-valued bifunctions,we introduce the notion of sequentially lower monotonity,which is strictly weaker than the lower semi-continuity of the second variables of the bifunctions.Then,we give a general version of vectorial Ekeland variational principle(briefly,denoted by EVP) for a system of equilibrium problems,where the sequentially lower monotone objective bifunction family is defined on products of sequentially lower complete spaces(concerning sequentially lower complete spaces,see Zhu et al(2013)),and taking values in a quasi-ordered locally convex space.Besides,the perturbation consists of a subset of the ordering cone and a family {p_i}_(i∈I) of negative functions satisfying for each i∈I,p_i(x_i,y_i) = 0 if and only if x_i=y_i.From the general version,we can deduce several particular equilibrium versions of EVP,which can be applied to show the existence of solutions for countable systems of equilibrium problems.In particular,we obtain a general existence result of solutions for countable systems of equilibrium problems in the setting of sequentially lower complete spaces.By weakening the compactness of domains and the lower semi-continuity of objective bifunctions,we extend and improve some known existence results of solutions for countable system of equilibrium problems in the setting of complete metric spaces(or Fréchet spaces).When the domains are non-compact,by using the theory of angelic spaces(see Floret(1980)),we generalize some existence results on solutions for countable systems of equilibrium problems by extending the framework from reflexive Banach spaces to the strong duals of weakly compactly generated spaces.