On equilibrium multivalued problems in the general model with the Boolean-valued bifunction

Abstract

In this paper, we establish some sufficient conditions for the existence of a solution of the equilibrium problem. The problem is understood in general form: find x ∈ E such that Φ(x, y) (−D\{0}) c for all y ∈ E; and some of its variants, where is a Boolean-valued bifunction and Φ is a multivalued mapping with values in vector space without topological structure. We use this result to show that (C, )-saddle point exists for multi-valued mappings. The results are established by combining the concepts of cyclic quasi-monotonicity and "algebraic" semicontinuity. Our results are interesting and refreshing because we do not need to use the convex hypothesis.1. Introduction. Let E and F be two non-empty sets and given a bifunction Φ : E × F → R. The problem which consists of finding an element x ∈ E satisfying Φ(x, y) ≥ 0 for all y ∈ F(1)has been studied previously in the works of Ky Fan [6], Muu and Oettli [18], Blum and Oettli [2] and others, with the term "equilibrium problem" (in short, EP Φ ). This problem and its related topics play important roles in many different areas of sciences, such as electrical circuits, economics, finance, game theory, constrained optimization, traffic network equilibrium problem, etc (see, e.g., [1,5,15,17,[19][20][21]).The variations of the problem (EP Φ ) are often considered with convex assumptions in a topologically structured environment, such as, in [15], the authors study the second-order optimality conditions for a class of the circular conic optimization problem, in that, the explicit expressions of the tangent cone and the second-order tangent set are established to discuss the minimal problems. In the work [16], the authors study the solubilities of optimization problems associated with the secondorder cone.The study of EP Φ has been extended from scalar to the vector case (for instance, [3,20] and the references therein). To formulate the problem (1), let V be a real