Existence conditions for bounded solutions of weakly perturbed linear impulsive systems. (English) Zbl 1230.34015

Boichuk, Alexander; Langerová, Martina; Škoríková, Jaroslava

Abstr. Appl. Anal. 2011, Article ID 792689, 13 p. (2011).

Summary: Weakly perturbed linear nonhomogeneous impulsive systems of the form \[ \dot{x} = A(t)x + \varepsilon A_1(t)x + f(t),\quad t \in \mathbb R,\;t \notin \mathcal T := \{\tau_i\}_{\mathbb Z}, \]
\[ \Delta x|_{t=\tau_i} = \gamma_i + \varepsilon A_{1i}x(\tau_i-),\quad \tau_i \in \mathcal T \subset \mathbb R,\;\gamma_i \in \mathbb R^n,\;i \in \mathbb Z \] are considered. Under the assumption that the generating system (for \(\varepsilon = 0\)) 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 the small parameter \(\varepsilon\) with finitely many terms with negative powers of \(\varepsilon\), and we suggest an algorithm for the construction of these solutions.

MSC:

34A37	Ordinary differential equations with impulses
34C11	Growth and boundedness of solutions to ordinary differential equations
34E05	Asymptotic expansions of solutions to ordinary differential equations
34D10	Perturbations of ordinary differential equations

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