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

• Autorzy: Hông Thái Nguyêñ
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytski{\u\i}, Uryson, etc.)
• 35R70: Partial differential equations with multivalued right-hand sides
• 47H04: Set-valued operators
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 54C60: Set-valued maps
• 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections
• 54C65: Selections

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2004
• Tom: 163
• Numer: 1
• Strony: 1-19

Daty

wydano

2004

Twórcy

autor

Hông Thái Nguyêñ

• Institute of Mathematics, Szczecin University, Wielkopolska 15, 70-451 Szczecin, Poland

Identyfikatory

DOI

10.4064/sm163-1-1