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Abstract
We apply some continuous selection theorems to establish coincidence theorems for a family of multimaps under various conditions. Then we apply these coincidence theorems to study the equilibrium problem with m families of players and 2m families of constraints on strategy sets. We establish the existence theorems of equilibria of this problem and existence theorem of equilibria of abstract economics with two families of players.
Highlights
For multimaps F : X Y and S : Y X, a point (x, y) ∈ X × Y is called a coincidence point of F and S if y ∈ F(x) and x ∈ S(y)
For the special case of our problem, if each of the families contain one player, we find that the strategy of each player is subject a constraint which is a function of strategies of the other players, and for each k ∈ I, where I is the index set of players, the strategies combination of the players other than the kth player is a function of strategy of kth player
The abstract economic problem, we studied in this paper, is different from the abstract economic studied in the literature
Summary
For multimaps F : X Y and S : Y X, a point (x, y) ∈ X × Y is called a coincidence point of F and S if y ∈ F(x) and x ∈ S(y). We apply the continuous selection theorem of Horvath [10] and the fixed-point theorem of Park [17] to derive the coincidence theorems for two families of multimaps. We will employ our results on coincidence theorems for two families of multimaps to consider the equilibrium problem with m families of players and 2m families of constraints on the strategy sets introduced by Lin et al [14]. For the special case of our problem, if each of the families contain one player, we find that the strategy of each player is subject a constraint which is a function of strategies of the other players, and for each k ∈ I, where I is the index set of players, the strategies combination of the players other than the kth player is a function of strategy of kth player. We want to establish the existence theorems of equilibria of constrained equilibrium problems with m families of players and 2m families of constraints and existence theorem of equilibria of abstract economic with two families of players
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