This paper studies some existence results for GENERALIZED e-vector EQUILIBRIUM PROBLEMS and GENERALIZED e-vector variational inequalities. The existence results for solutions are derived by using the celebrated KKM theorem. The results achieved in this paper generalize and improve the works of many authors in references.

The purpose of this paper is to study the solvability for a class of GENERALIZED strong vector EQUILIBRIUM-like PROBLEMS in reflexive Banach spaces. Firstly, utilizing Brouwer’s fixed point theorem, we prove the solvability for this class of GENERALIZED strong vector EQUILIBRIUM-like PROBLEMS without monotonicity assumption. Secondly, we introduce the new concept of pseudomonotonicity for vector set-valued mapping and prove the solvability for this class of GENERALIZED strong vector EQUILIBRIUM-like PROBLEMS for pseudomonotone vector set-valued mapping by using Fan’s lemma and Nadler’s theorem. Our results extend and improve the corresponding results in this direction.

In this paper, first a brief history of EQUILIBRIUM PROBLEMS(EP) and GENERALIZED implicit vector EQUILIBRIUM PROBLEMS(GIVEP) are given. Then some existence theorems for GIVEP are presented, also some suitable conditions in order the solution set of GIVEP is compact and convex for set-valued mappings whose are a subset of the cartesian product of Hausdorff topological vector space and their range is a subset of a topological space values (not necessarily locally convex or a topological vector space). In almost all of published results for GIVEP the setvalued mappings are considered from a topological vector space (locally convex topological vector space) to a topological vector space while in this paper the range of the set-valued mappings are a subsets of a topological spaces. As applications of our results, we derive some suitable conditions for existing a normalized Nash EQUILIBRIUM problem when the number of players are finite and the abstract case, that is infinite players. Finally, a numerical result, as an application of the main results, is given. The method used for proving the existence theorems is based on finite intersection theorems and Ky-Fan’, s theorem. The results of this paper, can be considered as suitable generalizations of the published paper in this area.

In this paper, we introduce two iterative schemes for finding a common solution of a GENERALIZED vector EQUILIBRIUM problem and relatively nonexpansive mappings in a real Banach space. We study the strong and weak convergence of the sequences generated by the proposed iterative schemes. The results presented in this paper are the supplement, extension, and generalization of the previously known results in this area.