Boundedness and almost periodicity of solutions of linear differential systems

Abstract

Abstract In this paper, we study the following linear differential system (1) x ′ ( t ) = A ( t ) x ( t ) , x ( t ) ∈ ℝ n , t ∈ ℝ , $${{x}^{\prime }}(t)=A(t)x(t),\,\,\,\,x(t)\in {{\mathbb{R}}^{n}},\quad t\in \mathbb{R},$$ where t ↦ A(t) is a matrix valued almost periodic function. We prove that if all the solutions of the above system are almost periodic, there exists an almost periodic function b : R → R n such that the following differential equation (2) x ′ ( t ) = A ( t ) x ( t ) + b ( t ) , x ( t ) ∈ ℝ n , t ∈ ℝ $${{x}^{\prime }}(t)=A(t)x(t)+b(t),\,\,\,\,x(t)\in {{\mathbb{R}}^{n}},\quad t\in \mathbb{R}$$ has no bounded solution. In particular, if for each almost periodic function b there exists a bounded solution to (2), there exists at least one solution for (1) that is not almost periodic.