Note that all these kinds of deÿnitions are motivated by the Fan-Browder ÿxed point theorem (e.g., see [10,20]), that is, in order to give the existence theorems of maximal elements, we can reduce the existence problem of maximal elements into the equivalent existence problem of ÿxed points of corresponding mappings. This idea motivates us to give a general deÿnition of Ky Fan (in short, F) set-valued mappings which enables us to use ÿxed point theorems of the Fan-Browder type to prove the existence of maximal elements. Moreover, our F-majorized mappings unify all other kinds of mappings mentioned above. It is our objective in this paper to give a uniÿed approach to study the existence of maximal elements and equilibria for qualitative games, generalized games in topological vector spaces and locally convex topological vector spaces.
In this paper, we shall ÿrst introduce the notions of correspondence of class F, F-majorant of at x and F-majorized correspondences which include corresponding deÿnitions of Ding and Tan [15], Ding and Tarafdar [16], Tan and Yuan [49], Tuclea [54] as special cases. We then discuss some ÿxed point theorems of the Fan-Browder type. These results improve corresponding results of Ben-El-Mechaiekh et al. [5], Border [8], Ding and Tan [15], Dugundji and Granas [18], Tarafdar [51], etc. By employing ÿxed point theorems, an existence theorem of maximal elements for F-majorized mappings is obtained which generalizes the corresponding results of Borglin and Keiding [9], Ding and Tan [15], Tarafdar [51], Toussaint [53], Tulcea [54, 55]
and others. As applications of our existence theorem for maximal elements, we prove equilibrium existence theorems for a non-compact one-person game and for a noncompact qualitative game with an inÿnite number of players and with F majorized correspondences. The latter result is applied to obtain an equilibrium existence theorem for a non-compact generalized game with an inÿnite number of players and with F correspondences. Finally, by introducing the concept 'approximate equilibria', the existence theorems of equilibria for one-person games and generalized games are obtained when constraint mappings are lower semicontinuous instead of having open inverse values in locally convex spaces. Our results unify and improve corresponding results in the literature, e.g., see [1-16, 22-26, 28, 29, 34-57, 59]. Now we need some preliminaries and deÿnitions. Throughout this paper, all topological spaces are assumed to be Hausdor unless otherwise speciÿed. The set of all real numbers is denoted by R. Let A be a subset of a topological space X . We shall denote by 2 A the family of all subsets of A, by F(A) the family of all non-empty ÿnite subsets of A, by int x (A) the interior of A in X and by cl X (A) the closure of A in X . The subset A is said to be compactly open in X if for each non-empty compact subset C of X , A ∩ C is open in C. If A is a subset of a vector space, we shall denote by co A the convex hull of A.