Existence Conditions for Bounded Solutions of Weakly Perturbed Linear ImpulsiveSystems

Abstract

The weakly perturbed linear nonhomogeneous impulsive systems in the form $\dot{x}=A\left(t\right)x\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\epsilon A_1\left(t\right)x\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}f\left(t\right),\mathrm{\hspace{0.17em}\hspace{0.17em}}t\in \mathbb{R},\mathrm{\hspace{0.17em}\hspace{0.17em}}t\notin \mathcal{T}:=\{{\tau }_i{\}}_{\mathbb{Z}},\Delta x{|}_{t={\tau }_i}={\gamma }_i+\epsilon A_{1i}x\left({\tau }_i-\right),\mathrm{\hspace{0.17em}\hspace{0.17em}}{\tau }_i\in \mathcal{T}\subset \mathbb{R},\mathrm{\hspace{0.17em}\hspace{0.17em}}{\gamma }_i\in {\mathbb{R}}^n$, and $i\in \mathbb{Z}$ are considered. Under the assumption that the generating system (for $\epsilon =0$) does not have solutions bounded on the entire real axis for some nonhomogeneities and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these systems bounded on the entire real axis in the form of a Laurent series in powers of small parameter $\epsilon$ with finitely many terms with negative powers of $\epsilon$, and we suggest an algorithm of construction of these solutions.