MMN-1516
A new criterion for the roughness of exponential dichotomy on R
Abstract
It is well known that system $x'=A(t)x$ possesses exponential dichotomy on $\mathbb{R}^+$ iff so does perturbed system $x'=(A(t)+B(t))x$, where perturbation $B(t)$ is ``small'', that is $\lim_{t\to+\infty}\left\|B(t)\right\|=0$. However, it is also known that corresponding statement for exponential dichotomy on $\mathbb{R}$ fails. In this work we show that under additional hypothesis of commutativity of $A(s)$ and $B(t)$ for all $s,t\in\mathbb{R}$ it can indeed be shown that dichotomy on $\mathbb{R}$ is unaffected by ``small'' perturbations (while ``small'' now has meaning of ``being in $L^1$ class''). We also provide some examples and particular corollaries of theory to show that hypotheses are natural and apply to wide range of problems.
Vol. 16 (2015), No. 2, pp. 987-994
DOI: 10.18514/MMN.2015.1516