Solvability of differential inclusions in a fixed set of functions. (Q1432454)

Consider the initial value problem for a functional-differential inclusion in a finite-dimensional space \[ \frac{dx}{dt}(t)\in F(t,x(\cdot),x(\cdot)),\;x(t_0)=x_0\in\mathbb R^n. \] The author derives sufficient conditions for the existence of at least one absolutely continuous solution belonging to a given closed set of continuum solutions. The proofs are based on the fixed-point theorem due to Eilenberg-Montgomery.