On the question of the existence of continuous branches of multivalued mappings with nonconvex images in spaces of summable functions

A. I. Bulgakov

Sbornik. Mathematics, Tome 64 (1989) no. 1, pp. 295-303

Résumé

Let $B$ be a Banach space with norm $\|\cdot\|$, and let $(E,\mathfrak M)$ be a compact topological space with $\sigma$-algebra of measurable sets $\mathfrak M$ on which a nonnegative regular Borel measure $\mu$ is given. Further, let $L_1(E,B)$ be the Banach space of Bochner-integrable functions $u\colon E\to B$, with the norm $\|u\|_{L_1(E,B)}=\int_E\|u(t)\|\,d\mu$, and let $\Phi\colon K\to2^{L_1(E,B)}$ be a multivalued mapping and $P\colon K\to L_1(E,B)$ a single-valued mapping, where $K$ is a compact topological space. Under certain assumptions it is proved that for any $\varepsilon>0$ there exists a continuous mapping $g\colon K\to L_1(E,B)$ such that the following conditions hold for any $x\in K$: $g(x)\in\Phi(x)$, and $\|P(x)-g(x)\|_{L_1(E,B)}\rho_{L_1(E,B)}[P(x),\Phi(x)]+\varepsilon$, where $\rho_{L_1(E,B)}[\,\cdot\,{,}\,\cdot\,]$ is the distance in $L_1(E,B)$ from a point to a set. Bibliography: 11 titles.