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Quasi-variational inequalities and social equilibrium. (English) Zbl 0783.49005

Let \(E\) be a topological vector space (TVS) and \(X\) a nonempty subset of \(E\). A multivalued mapping \(F: X\to 2^ X\) and a function \(f: X\times X\to\mathbb{R}\) are usually assumed to enjoy the properties: (a) \(F\) is compact-convex-valued and upper hemi-continuous (in the sense of J.- P. Aubin and I. Ekeland [‘Applied nonlinear analysis’ (1984; Zbl 0641.47066)]), (b) \(x\mapsto f(x,y)\) is lower semi-continuous for each fixed \(y\in X\).
The problem is to find out a sufficient condition which guarantees the existence of a point \(x^*\in X\) such that (i) \(x^*\in F(x^*)\) and (ii) \(\sup_{y\in F(x^*)} f(x^*,y)\leq 0\).
Aubin-Ekeland solved this problem in the case that \(E\) is a Hilbert space and \(X\) is compact convex. J. X. Zhou and G. Chen [J. Math. Anal. Appl. 132, No. 1, 213-225 (1988; Zbl 0649.49008)] generalized this result to the case that \(E\) is a locally compact Hausdorff TVS. The main theorem of the present paper tells us that it is still possible to find out a way leading to the same goal even when \(E\) is a more general TVS and the compactness of \(X\) is dispensed with.
As an application of this main result, an existence theorem of the Cournot-Nash equilibria in \(n\)-person noncooperative game is provided in a general framework, where the set of feasible strategies of each player depends upon the choice of strategies of other players. (However I wonder if there is any essential difference between Theorem 3.1 and Theorem 4.1).
Reviewer: T.Maruyama (Tokyo)

MSC:

49J40 Variational inequalities
91A10 Noncooperative games
49J45 Methods involving semicontinuity and convergence; relaxation
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References:

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