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.

中文翻译：

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线性微分系统解的有界性和近似周期性

在本文中，我们研究以下线性微分系统 (1) X ‘ ( 吨 ) = 一种 ( 吨 ) X ( 吨 ) , X ( 吨 ) ∈ ℝ n , 吨 ∈ ℝ , $${{x}^{\prime }}(t)=A(t)x(t),\,\,\,\,x(t)\in {{\mathbb{R}}^{n }},\quad t\in \mathbb{R},$$ 在哪里吨↦一个(吨) 是矩阵值的近似周期函数。我们证明如果上述系统的所有解都是几乎周期的，则存在一个几乎周期的函数b: 回复→R n 使得下面的微分方程 (2) X ‘ ( 吨 ) = 一种 ( 吨 ) X ( 吨 ) + b ( 吨 ) , X ( 吨 ) ∈ ℝ n , 吨 ∈ ℝ $${{x}^{\prime }}(t)=A(t)x(t)+b(t),\,\,\,\,x(t)\in {{\mathbb{R }}^{n}},\quad t\in \mathbb{R}$$ 没有有界解。特别是，如果对于每个几乎周期性的函数b(2) 存在有界解， (1) 至少存在一个几乎不是周期性的解。