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

Hôǹg Thái Nguyêñ ; Dariusz Pączka

Banach Center Publications, Tome 83 (2008), p. 135-156 / Harvested from The Polish Digital Mathematics Library

Résumé

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).

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Publié le : 2008-01-01

Zbl 1138.49020

EUDML-ID : urn:eudml:doc:281936