The Clarke tangent and normal cones to decomposable sets in Lebesgue spaces

Abstract

Let $(T,Σ,μ)$ be a complete, $σ$-finite measure space, let $Z$ be a separable Banach space, and let $S : T \rightrightarrows Z$ be a measurable multifunction with nonempty closed values. For $1 \leq p\leq \infty,$ we consider $$\mathsf{Sel}_{p}(S):=\{z \in L^p(T,Z) : z(t) \in S(t) \hspace{0.1cm} \text{a.e.}\}.$$ We study whether the Clarke tangent cone to $\mathsf{Sel}_{p}(S)$ is obtained by taking $L^p$-selections of the pointwise Clarke tangent cones to the values of $S;$ namely, whether $ \operatorname{\widehat{\textbf{T}}}_{\mathsf{Sel}_{p}(S)}(x)=\{v \in L^p(T,Z) : v(t) \in \operatorname{\widehat{\textbf{T}}}_{S(t)}(x(t)) \hspace{0.1cm} \text{a.e.}\} $ holds for $x \in \mathsf{Sel}_{p}(S).$ The main result gives an affirmative answer for $1 \leq p < \infty$ under the additional assumption that $Z$ is reflexive. If $p=\infty,$ we prove $\widehat{\mathbf T}_{\mathrm{Sel}_{\infty}(S)}(x)\subset \{v\in L^\infty(T,Z): v(t)\in \widehat{\mathbf T}_{S(t)}(x(t))\ \mathrm{a.e.}\}.$other possible partial results are discussed. Consequently the corresponding assertions for Clarke normal cones follow. We derive applications to nonsmooth constrained optimization problems, Nemytskii operators and minimization of integral functional with decomposable constraints.