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Bounded solutions of linear perturbed differential equations in a Banach space. (English) Zbl 1164.34025
The linear weakly perturbed differential equation $$ \dot x=A(t)x+\varepsilon A_1(t)x+f(t) $$ in Banach spaces is studied for $t\in \Bbb R$ by supposing that $\dot x=A(t)x$ has exponential dichotomies on both semi-axes $\Bbb R_\pm $. Conditions on the existence of bounded solutions on $\Bbb R$ of the above weakly perturbed equation are established. Concrete examples are presented at the end of the paper.

34G10Linear ODE in abstract spaces
34C11Qualitative theory of solutions of ODE: growth, boundedness
34D09Dichotomy, trichotomy