Stable perturbations of linear differential equations generating a uniformly bounded group

V. V. Skazka<sup>ab

a</sup> Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^b Novosibirsk State University

Abstract: Stability problems for solutions of the differential equation $u'(t)=Au+\varepsilon B(t,u)$ in a Banach space are considered. It is assumed that for $\varepsilon=0$ this equation generates a uniformly bounded group of class $C_0$. Sufficient conditions on $B$ and $A$ are found under which the solutions of this equation are bounded for small $\varepsilon$. A linearization principle is proved for this equation under certain conditions on the operator $B$.
Bibliography: 9 titles.

Keywords: differential equations in a Banach space, stability of solutions.

UDC: 517.95

MSC: Primary 34G10; Secondary 34D05, 70K28

Received: 27.12.2016

DOI: 10.4213/sm8895

 English version: Sbornik: Mathematics, 2017, 208:8, 1246–1259

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