Stable perturbations of linear differential equations generating a uniformly bounded group

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 $B$.
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