Carathéodory convex selections of set-valued functions in Banach lattices

Abstract

Let $T$ be a measurable space, $X$ a Banach space while $Y$ a Banach lattice. We consider the class of "upper separated" set-valued functions $F\colon T\times X \rightarrow 2^{Y}$ and investigate the problem of the existence of Carathéodory type selection, that is, measurable in the first variable and order-convex in the second variable.