Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions

Hông Thái Nguyêñ

Studia Mathematica, Tome 162 (2004), p. 1-19 / Harvested from The Polish Digital Mathematics Library

Résumé

Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction $f: Ω × E → 2^{F}$, denote by $N_{f}(x)$ the set of all measurable selections of the multifunction $f(·,x(·)): Ω → 2^{F}$, s ↦ f(s,x(s)), for a function x: Ω → 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 (Nemytskiĭ) superposition operator $N_{f}$ mapping some open domain G ⊂ 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 Ω ⊂ ℝ².

Détail

Publié le : 2004-01-01

Zbl 1071.47049

EUDML-ID : urn:eudml:doc:285266