Almost-periodic and bounded solutions of Carathéodory differential inclusions

Abstract

The main purpose of this paper is two-fold: to find sufficient conditions for the existence of entirely bounded solutions of Carathéodory quasi-linear differential inclusions and to show that, if the coefficients are specially constant and the right-hand sides are additionally Lipschitz continuous (with a sufficiently small Lipschitz constant) and almost-periodic in time, then these solutions become almost-periodic as well. The almost-periodicity is understood in the sense of H. Weyl and, because of set-valued analysis, we introduce for the first time the appropriately generalized concept. The related methods, including the fixed-point theorem for a class of $\mathcal{J}$-maps in locally convex topological vector spaces, are developed here too. In the single-valued case, the obtained criteria generalize those of the other authors.