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.


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
Full Text: DOI


[1] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, Vyshcha Shkola, Kiev, Ukraine, 1974. · Zbl 0837.34003
[2] S. Schwabik, M. Tvrdý, and O. Vejvoda, Differential and Integral Equations, Academia, Prague, Czech Republic, 1979. · Zbl 0424.34014
[3] A. A. Boichuk, N. A. Perestyuk, and A. M. Samoilenko, “Periodic solutions of impulse differential systems in critical cases,” Differents. Uravn., vol. 27, no. 9, pp. 1516-1521, 1991. · Zbl 0778.34007
[4] R. J. Sacker, “The splitting index for linear differential systems,” Journal of Differential Equations, vol. 33, no. 3, pp. 368-405, 1979. · Zbl 0438.34008
[5] K. J. Palmer, “Exponential dichotomies and transversal homoclinic points,” Journal of Differential Equations, vol. 55, no. 2, pp. 225-256, 1984. · Zbl 0508.58035
[6] A. M. Samoilenko, A. A. Boichuk, and An. A. Boichuk, “Solutions, bounded on the whole axis, of linear weakly perturbed systems,” Ukrainian Mathematical Journal, vol. 54, no. 11, pp. 1517-1530, 2002. · Zbl 1041.34035
[7] A. A. Boichuk, “Solutions of weakly nonlinear differential equations bounded on the whole line,” Nonlinear Oscillations, vol. 2, no. 1, pp. 3-10, 1999. · Zbl 1035.34021
[8] A. Boichuk and A. Pokutnyi, “Bounded solutions of linear perturbed differential equations in a Banach space,” Tatra Mountains Mathematical Publications, vol. 39, pp. 1-12, 2007. · Zbl 1164.34025
[9] F. Battelli and M. Fe\vckan, “Chaos in singular impulsive O.D.E.,” Nonlinear Analysis, vol. 28, no. 4, pp. 655-671, 1997. · Zbl 0919.34016
[10] A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, Koninklijke Brill NV, Utrecht, Mass, USA, 2004. · Zbl 1083.47003
[11] A. A. Boichuk, J. Diblík, D. Khusainov, and M. Rů\vzi, “Fredholm’s boundary-value problems for differential systems with a single delay,” Nonlinear Analysis, vol. 72, no. 5, pp. 2251-2258, 2010. · Zbl 1190.34073
[12] A. A. Boichuk, J. Diblík, D. Khusainov, and M. Rů\vzi, “Boundary-value problems for weakly nonlinear delay differential systems,” Abstract and Applied Analysis, vol. 2011, Article ID 631412, 19 pages, 2011. · Zbl 1222.34075
[13] A. Boichuk, M. Langerová, and J. , “Solutions of linear impulsive differential systems bounded on the entire real axis,” Advances in Difference Equations, vol. 2010, Article ID 494379, 10 pages, 2010. · Zbl 1204.34040
[14] M. I. Vishik and L. A. Lyusternik, “The solution of some perturbation problems for matrices and selfadjoint differential equations,” Uspekhi Matematicheskikh Nauk, vol. 15, no. 3, pp. 3-80, 1960. · Zbl 0096.08702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.