Generalized gradients for locally Lipschitz integral functionals on non-$L^p$-type spaces of measurable functions
Tom 79 / 2008
Streszczenie
Let $(\Omega ,\mu )$ be a measure space, $E$ be an arbitrary separable Banach space, $E_{\omega ^{\ast }}^{\ast }$ be the dual equipped with the weak$^{\ast }$ topology, and $g:\Omega \times E\rightarrow {\mathbb{R}}$ be a Carathéodory function which is Lipschitz continuous on each ball of $E$ for almost all $s\in \Omega $. Put $G(x):=\int_{\Omega }g(s,x(s))d\mu(s)$. Consider the integral functional $G$ defined on some non-$% L^{p}$-type Banach space $X$ of measurable functions $x\colon \Omega \rightarrow E$. We present several general theorems on sufficient conditions under which any element $\gamma \in {X}^{\ast }$ of Clarke's generalized gradient (multivalued $C$-subgradient) $\partial _{C}G(x)$ has the representation $\gamma (v)=\int_{\Omega }\langle \zeta (s),v(s)\rangle d\mu ( s) \,(v\in {X})$ via some measurable function $\zeta \colon \Omega \rightarrow E_{w^{\ast }}^{\ast }$ of the associate space $X^{\prime } $ such that $\zeta (s)\in \partial _{C}g(s,x(s))$ for almost all $s\in \Omega $. Here, given a fixed $s\in \Omega $, $\partial _{C}g(s,u_{0})$ denotes Clarke's generalized gradient for the function $g(s,\cdot )$ at $% u_{0}\in E$. 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).
