Universal perturbations of linear differential equations

Abstract:

Let $X:[0,\infty )\to L(\mathbb {R}^n)$ be a fundamental solution of $x’=A( t)x$ with $X$ and $X^{-1}$ bounded on $[0,\infty )$. We prove that there exist arbitrary small matrix functions $B:[0,\infty )\to L(\mathbb {R}^n)$ with limit $0$ as $t\to \infty$ such that $y’=(A(t)+B(t))y$ has solutions with $y([0,\infty ))$ dense in $\mathbb {R}^n$.