Quasi-variational inequality problems with non-compact valued constraint maps

Coerciveness condition for quasi-equilibrium problems

“…The previous result is strongly related to Theorem 5 in [3]. However, it is important to notice that in the set of assumptions A1 the constraint set-valued maps are not closed.…”

“…In the spirit of Proposition 1 in [3], we will show that the second condition of UCC holds for each element of QEP(f, K) under generalized monotonicity. Proposition 3.1.…”

“…Remark 1. Lemma 2.3 is a slight refinement of Lemma 1 in [3], mainly because it drops two assumptions: the values of T need not to be closed, and the values of T V need not to have non-empty interior.…”

“…1. The previous result is not a consequence of Theorem 1 in [3], because T is properly quasi-monotone and K is not closed. Theorem 3 in [3] proposes an existence result under quasi-monotonicity, that means for all (x, x * ) and (y, y * ) in the graph of T the following implication holds…”

“…Our aim in this work is to provide some existence results for (QEP), under a coerciveness condition which is inspired from [3]. In Section 2 we present basic and classical notions on generalized convexity, generalized monotonicity, continuity for set-valued maps and some results.…”