Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions
Volume 163 / 2004
Abstract
Let ${\mit \Omega }$ be a measure space, and $E$, $F$ be separable Banach spaces. Given a multifunction $f:{\mit \Omega }\times E\to 2^F$, denote by $N_f(x)$ the set of all measurable selections of the multifunction $f(\cdot ,x(\cdot )): {\mit \Omega }\to 2^{F}$, $s\mapsto f(s,x(s))$, for a function $x: {\mit \Omega }\to E$. First, we obtain new theorems on $H$-upper/$H$-lower//lower semicontinuity (without assuming any conditions on the growth of the generating multifunction $f(s,u)$ with respect to $u$) for the multivalued (Nemytski{ĭ}) superposition operator $N_f$ mapping some open domain $G\subset X$ into $2^Y$, where $X$ and $Y$ are Köthe–Bochner spaces (including Orlicz–Bochner spaces) of functions taking values in Banach spaces $E$ and $F$ respectively. Second, we obtain a new theorem on the existence of continuous selections for $N_f$ taking nonconvex values in non-$L_p$-type spaces. Third, applying this selection theorem, we establish a new existence result for the Dirichlet elliptic inclusion in Orlicz spaces involving a vector Laplacian and a lower semicontinuous nonconvex-valued right-hand side, subject to Dirichlet boundary conditions on a domain ${\mit \Omega }\subset {\mathbb R}^2$.
