DISCONTINUOUS GENERALIZED QUASI-VARIATIONAL INEQUALITIES WITH APPLICATION TO FIXED POINTS

Abstract

We consider the following generalized quasi-variational inequality problem introduced in [7]: given a real normed space $X$ with topological dual $X^*$, two sets $C,D \subseteq X$ and two multifunctions $S: C \to 2^D$ and $T: C \to 2^{X^*}$, find $(\hat x,\hat\varphi) \in C \times X^*$ such that $$\hat x \in S(\hat x), \quad\hat\varphi \in T(\hat x) \quad{\rm and} \quad \langle\hat\varphi,\hat x-y\rangle \le 0 \quad{\rm for\ all}\quad y\in S(\hat x).$$ We prove an existence theorem where $T$ is not assumed to have any continuity or monotonicity property, improving some aspects of the main result of [7]. In particular, the coercivity assumption is meaningfully weakened. As an application, we prove a theorem of the alternative for the fixed points of a Hausdorff lower semicontinuous multifunction. In particular, we obtain sufficient conditions for the existence of a fixed point which belongs to the relative boundary of the corresponding value.