Solution sets of multivalued Sturm-Liouville problems in Banach spaces

In the paper we present an abstract existence theorem for a boundary value problem in an arbitrary Banach space $X$ of the form $\dot x = f(t,x)$, $t\in I=[a,b]$, $x\in S$, where $f\colon I\times X\to X$ is a continuous mapping, and $S$ is a subset of $C(I,X)$, is proved. The conditions are formulated in terms of a ``diagonal'' representation of the corresponding ``integral'' operator, which is assumed to be densifying in the first variable with respect to a monotone and regular measure of noncompactness.