An Order-Theoretical Multi-Valued Fixed Point Approach to Quasi-Variational Inclusions with Bifunctions

Abstract

We present an order-theoretical fixed point theorem for increasing multivalued operators suitable for the method of sub-supersolutions and its application to the following multivalued quasi-variational inclusion: Let $\Omega \subset \mathbb R^N$ be a bounded Lipschitz domain and $W = W_0^{1,p}(\Omega)$. Find $u\in W$ such that for some measurable selection $\eta$ of $f(\cdot,u,u)$ it holds \begin{equation*} \langle Eu,w-u\rangle + \int_\Omega \eta(w-u) + K(w,u) - K(u,u) \geq 0\quad\text{for all }w\in W, \end{equation*} where $E\colon W \to W^\ast$ is an elliptic Leray-Lions operator of divergence form, $f\colon \Omega \times \mathbb R\times \mathbb R \to \mathcal P(\mathbb R)$ is a multivalued bifunction being upper semicontinuous in the second and decreasing in the third argument, and $K(\cdot,u)$ is a convex functional for each $u\in W$. Under weak assumptions on the data we will prove that there are smallest and greatest solutions between each pair of appropriately defined sub-supersolutions.