On the Existence of Partially Irregular Almost Periodic Solutions of Linear Nonhomogeneous Differential Systems in a Critical Nonresonance Case

Abstract

where the n × n matrix function A(t) and the n-vector function φ(t) are almost periodic. Almost periodic solutions of (1) whose frequency module coincides with that of the right-hand side were studied in numerous papers (e.g., see [1–4] etc.). The existence of irregular almost periodic solutions, that is, solutions with zero intersection of the frequency modules of the solution and of system (1) itself, were analyzed in [5]. In particular, it was shown that such solutions can be only quasiperiodic. It is of interest to consider partially irregular solutions of system (1), that is, solutions such that the frequency module of the solution does not contain some submodule of the right-hand side. Throughout the following, we assume that the frequency modules of the coefficient matrix and the driving force are nontrivial and have the zero intersection