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Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

Author

Listed:
  • Alexander Boichuk
  • Martina Langerová
  • Jaroslava Škoríková

Abstract

The weakly perturbed linear nonhomogeneous impulsive systems in the form x.=A(t)x + εA1(t)x + f(t), t∈ℝ, t∉𝒯:={τi} ℤ, Δx|t=τi=γi+εA1ix(τi-), τi∈𝒯⊂ℝ, γi∈ℝn, 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.

Suggested Citation

  • Alexander Boichuk & Martina Langerová & Jaroslava Škoríková, 2011. "Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    • Handle: RePEc:wly:jnlaaa:v:2011:y:2011:i:1:n:792689
    DOI: 10.1155/2011/792689
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    References listed on IDEAS

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    1. A. Boichuk & J. Diblík & D. Khusainov & M. Růžičková, 2011. "Boundary‐Value Problems for Weakly Nonlinear Delay Differential Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    2. A. Boichuk & J. Diblík & D. Khusainov & M. Růžičková, 2011. "Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-19, June.
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