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

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.

\[ \Delta x|_{t=\tau_i} = \gamma_i + \varepsilon A_{1i}x(\tau_i-),\quad \tau_i \in \mathcal T \subset \mathbb R,\;\gamma_i \in \mathbb R^n,\;i \in \mathbb Z \] 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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\textit{A. Boichuk} et al., Abstr. Appl. Anal. 2011, Article ID 792689, 13 p. (2011; Zbl 1230.34015)

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### References:

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