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Perturbations of asymptotically stable differential systems. (English) Zbl 0568.34035

We consider systems of differential equations of the form (1) ${x}^{\text{'}}=f\left(t,x\right)$, for $t\in \left[a,\infty \right)$, x in some domain $D\subset {ℝ}^{n}$ and $f\in {C}^{1}\left(\left[a,\infty \right)×D\right)$ (a̠ a fixed real number). We assume that the solution $x\left(t,{t}_{0},{x}_{0}\right)$ of (1) defined for $t\ge a$ satisfies $|x\left(t,{t}_{0},{x}_{0}\right)|\le c|{x}_{0}|h\left(t\right)h{\left({t}_{0}\right)}^{-1}$ $\left(t\ge {t}_{0}\ge a\right)$ for ${x}_{0}$ small enough, for some constant $c>0$ and h a continuous positive function defined in [a,$\infty \right)$. We give conditions for the perturbed system $\left(2\right)\phantom{\rule{1.em}{0ex}}{y}^{\text{'}}=f\left(t,y\right)+g\left(t,y\right)$ (g$\in C\left(\left[a,\infty \right)×D\right)\right)$ to have the same type of stability as (1).

Since the paper appeared, we have obtained further general results about asymptotic integration for perturbed systems (2). Moreover, the corresponding results for linear systems give new insights about Levinson’s theorem on asymptotic integration. In this way using these systems (which we call h-systems) we get a uniform treatment for the concept of stability and we also verify that several classical theorems of stability take a precise form by the asymptotic formulae.

MSC:
 34D10 Stability perturbations of ODE 34C11 Qualitative theory of solutions of ODE: growth, boundedness