Existence of solutions for systems of generalized quasi‐equilibrium problems

Abstract

In this paper, first of all, we consider a generalized game in choice form with 2 constraints and its corresponding equilibrium in choice. We assert new conditions under which the equilibrium in choice exists. As a consequence, we establish the existence of the equilibrium for generalized abstract economies.Then, we apply the obtained theorems to prove the existence of solutions for systems of quasi-equilibrium problems. We do this by considering new hypotheses for the properties of the involved correspondences. This approach leads us to results which differ a lot from the ones existing in literature. KEYWORDS equilibrium in choice, generalized game in choice form, quasi-equilibrium problem Math Meth Appl Sci. 2018;41:803-817.wileyonlinelibrary.com/journal/mma 804 PATRICHE the approached topic. New techniques of proof are developed by defining new correspondences which are requested to verify the local intersection property. Further, a selection lemma and the Brouwer fixed-point theorem are appealed. Two versions of equilibrium, slightly different, are discussed: weak equilibrium and strong equilibrium. As direct applications, we prove that the equilibrium exists for generalized abstract economies in which the preference correspondences do not have convex values, and their upper or lower sections do not verify topological properties. This study is a continuation of the author's research work, concerning the existence of equilibrium, as it can be seen, for instance, in Hervés-Beloso and Patriche. [10][11][12] Secondly, we study the existence of the solutions for some systems of vector quasi-equilibrium problems. We emphasize the importance of research studies on this topic, justified by the fact that it is, in fact, a unified model of other several problems. We can mention, here, vector variational inequalities, vector optimization problems, or Debreu-type equilibrium problems. For further relevant information, the reader is referred to the following recent publications available in our bibliography. [13][14][15][16][17][18][19][20][21][22][23][24][25][26] We define the notion of "weak solution," and we show its existence by considering an associate generalized abstract economy model which admits weak equilibrium. In this way, we obtain results which are totally new in literature, and here, we refer to the hypotheses, as well as to the tools of demonstration. We show that very slight continuity properties of the involved correspondences can also lead to the existence of the solutions for various classes of equilibrium problems.The rest of the paper is organized as follows: In Section 2, the systems of vector quasi-equilibrium problems are presented, and some new definitions for correspondences are introduced. Section 3 contains results concerning the existence of equilibrium. In Section 4, the existence of solutions for systems of vector quasi-equilibrium problems is proved. The concluding remarks are presented in Section 5.