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

Hông Thái Nguyêñ. "Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions." Studia Mathematica 163.1 (2004): 1-19. <http://eudml.org/doc/285266>.

@article{HôngTháiNguyêñ2004,
abstract = {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.},
author = {Hông Thái Nguyêñ},
journal = {Studia Mathematica},
keywords = {Dirichlet elliptic inclusions; multivalued Nemytskiĭ superposition operator; multivalued locally defined operator; existence of continuous selections; Köthe-Bochner spaces},
language = {eng},
number = {1},
pages = {1-19},
title = {Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions},
url = {http://eudml.org/doc/285266},
volume = {163},
year = {2004},
}

TY - JOUR
AU - Hông Thái Nguyêñ
TI - Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions
JO - Studia Mathematica
PY - 2004
VL - 163
IS - 1
SP - 1
EP - 19
AB - 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.
LA - eng
KW - Dirichlet elliptic inclusions; multivalued Nemytskiĭ superposition operator; multivalued locally defined operator; existence of continuous selections; Köthe-Bochner spaces
UR - http://eudml.org/doc/285266
ER -