Generalized Quasi-Variational Inequalities in Infinite-Dimensional Normed Spaces

A result related to Ricceri's conjecture on generalized quasi-variational inequalities

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We obtain an existence theorem for generalized quasi-variational inequalities in infinite-dimensional normed spaces which improves some aspects of a recent result by P. Cubiotti [5], and gives a partial affirmative answer to a conjecture formulated by B. Ricceri [8].

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“…We refer to [5] and [8] for detailed discussions on sufficient conditions for generalized quasi-variational inequalities to have nonempty solution sets (the existence theorems). In [5], various applications of such problems were discussed. A new application to control theory has been recently given in [4].…”

“…Conjecture (B. Ricceri [8]; see also [5]). Assume that: F satisfies assumptions (A1) and (A2), G is lower semicontinuous, with closed graph and, for each x P X, Gx is a convex set which has nonempty interior in the affine hull of X.…”

“…One of us [5,Theorem 3.1] has proved that the conjecture is true under the following additional assumptions:…”

“…For proving this useful fact one can argue as with Proposition 2.4 in [5], where a reference to [10] was made. Here we give a simple direct proof.…”