Abstract and Applied Analysis
Volume 2011 (2011), Article ID 792689, 13 pages
http://dx.doi.org/10.1155/2011/792689
Research Article

Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

Alexander Boichuk,1 Martina Langerová,2 and Jaroslava Škoríková2

1Institute of Mathematics, National Academy of Science of Ukraine, 01601 Kyiv, Ukraine
2Department of Mathematics, University of Žilina, 01026 Žilina, Slovakia

Received 31 December 2010; Revised 1 July 2011; Accepted 1 July 2011

Academic Editor: Josef Diblík

Copyright © 2011 Alexander Boichuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The weakly perturbed linear nonhomogeneous impulsive systems in the form ̇ 𝑥 = 𝐴 ( 𝑡 ) 𝑥 + 𝜀 𝐴 1 ( 𝑡 ) 𝑥 + 𝑓 ( 𝑡 ) , 𝑡 ∈ ℝ , 𝑡 ∉ 𝒯 ∶ = { 𝜏 𝑖 } ℤ , Δ 𝑥 | 𝑡 = 𝜏 𝑖 = 𝛾 𝑖 + 𝜀 𝐴 1 𝑖 𝑥 ( 𝜏 𝑖 − ) , 𝜏 𝑖 ∈ 𝒯 ⊂ ℝ , 𝛾 𝑖 ∈ ℝ 𝑛 , and 𝑖 ∈ ℤ are considered. Under the assumption that the generating system (for 𝜀 = 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 𝜀 with finitely many terms with negative powers of 𝜀 , and we suggest an algorithm of construction of these solutions.