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

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.
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