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

Abstract

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 Frechet 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.