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An alternate version of a variational inequality. (English) Zbl 1034.49005

From the text: “The purpose of this paper is to give some other alternate version of a result, concerning a variational inequality due to G. Allen [J. Math. Anal. Appl. 58, 1–10 (1977; Zbl 0383.49005)].

The following theorem is the main result of this paper.

Theorem. Let X be a closed nonempty subset of a locally convex semi-reflexive topological vector space E and let f:X×X be a mapping such that:

(1) for each fixed yX, f(·,y):X is weakly usc on X.

(2) there exists a real c such that

(i) for each xX and t<c, the set {yX:f(x,y)t} is convex,

(ii) for each xX, f(x,x)c,

(iii) for a particular y 0 X, the set {xX:f,y 0 )c} is a bounded subset of E.

Then there exists an x 0 X such that f(x 0 ,y)c for all yX”.

49J40Variational methods including variational inequalities
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)