Generalized gradients for locally Lipschitz integral functionals on non-$L^p$-type spaces of measurable functions

• Autorzy: Hôǹg Thái Nguyêñ , Dariusz Pączka
• Języki publikacji: EN

Abstrakty

EN

Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, $E*_{ω*}$ be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put $G(x): = ∫_{Ω} g(s,x(s))dμ(s)$. Consider the integral functional G defined on some non-$L^{p}$-type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke's generalized gradient (multivalued C-subgradient) $∂_{C}G(x)$ has the representation $γ(v) = ∫_{Ω} ⟨ζ(s),v(s)⟩dμ(s) (v ∈ X)$ via some measurable function $ζ: Ω → E*_{w*}$ of the associate space X' such that $ζ(s) ∈ ∂_{C}g(s,x(s))$ for almost all s ∈ Ω. Here, given a fixed s ∈ Ω, $∂_{C}g(s,u₀)$ denotes Clarke's generalized gradient for the function g(s,·) at u₀ ∈ E. What concerning X, we suppose that it is either a so-called non-solid Banach M-space (in particular, non-solid generalized Orlicz space) or Köthe-Bochner space (solid space).

Kategorie tematyczne

• 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytski{\u\i}, Uryson, etc.)
• 35R70: Partial differential equations with multivalued right-hand sides
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 58C20: Differentiation theory (Gateaux, Fr\'echet, etc.)
• 49J52: Nonsmooth analysis

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2008
• Tom: 79
• Numer: 1
• Strony: 135-156

Daty

wydano

2008

Twórcy

autor

Hôǹg Thái Nguyêñ

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

autor

Dariusz Pączka

• Institute of Mathematics, Szczecin University of Technology, Al. Piastów 48/49, 70-310 Szczecin, Poland

Identyfikatory

DOI

10.4064/bc79-0-11