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

Hôǹg Thái Nguyêñ; Dariusz Pączka

Banach Center Publications (2008)

- Volume: 79, Issue: 1, page 135-156
- ISSN: 0137-6934

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topHôǹg Thái Nguyêñ, and Dariusz Pączka. "Generalized gradients for locally Lipschitz integral functionals on non-$L^p$-type spaces of measurable functions." Banach Center Publications 79.1 (2008): 135-156. <http://eudml.org/doc/281936>.

@article{HôǹgTháiNguyêñ2008,

abstract = {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).},

author = {Hôǹg Thái Nguyêñ, Dariusz Pączka},

journal = {Banach Center Publications},

keywords = {non-smooth analysis; generalized gradient; mulitivalued -subgradient; locally Lipschitz integral functional; Banach -space; Banach -module; non-solid generalized Orlicz space; Banach lattice; Köthe-Bochner space; Orlicz space; Dirichlet differential inclusion involving growth-exponential-type nonlinearity},

language = {eng},

number = {1},

pages = {135-156},

title = {Generalized gradients for locally Lipschitz integral functionals on non-$L^p$-type spaces of measurable functions},

url = {http://eudml.org/doc/281936},

volume = {79},

year = {2008},

}

TY - JOUR

AU - Hôǹg Thái Nguyêñ

AU - Dariusz Pączka

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

JO - Banach Center Publications

PY - 2008

VL - 79

IS - 1

SP - 135

EP - 156

AB - 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).

LA - eng

KW - non-smooth analysis; generalized gradient; mulitivalued -subgradient; locally Lipschitz integral functional; Banach -space; Banach -module; non-solid generalized Orlicz space; Banach lattice; Köthe-Bochner space; Orlicz space; Dirichlet differential inclusion involving growth-exponential-type nonlinearity

UR - http://eudml.org/doc/281936

ER -

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