Abstract
The weakly perturbed linear nonhomogeneous impulsive systems in the form , 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.
Highlights
In this contribution we study the problem of existence and construction of solutions of weakly perturbed linear differential systems with impulsive action bounded on the entire real axis
Δx|t τi γi, τi ∈ T, i ∈ Z, where A ∈ BCT R is an n × n matrix of functions, f ∈ BCT R is an n × 1 vector function, BCT R is the Banach space of real vector functions bounded on R and left-continuous for t ∈ R with discontinuities of the first kind at t ∈ T : {τi}Z with the norm: x BCT R : supt∈R x t, γi are n-dimensional column constant vectors: γi ∈ Rn; · · · < τ−2 < τ−1 < τ0 0 < τ1 < τ2 < · · ·, and Δx|t τi : x τi − x τi−
The solution x t of the system 2.1 is sought in the Banach space of ndimensional bounded on R and piecewise continuously differentiable vector functions with discontinuities of the first kind at t ∈ T : x ∈ BCT1 R
Summary
In this contribution we study the problem of existence and construction of solutions of weakly perturbed linear differential systems with impulsive action bounded on the entire real axis. The application of the theory of differential systems with impulsive action developed in 1–3 , the well-known results on the splitting index by Sacker 4 and by Palmer 5 on the Fredholm property of bounded solutions of linear systems of ordinary differential equations 6–9 , the theory of pseudoinverse matrices 10 and results obtained in analyzing boundary-value problems for ordinary differential equations see 10–12 , enables us to obtain existence conditions and to propose an algorithm for the construction of solutions bounded on the entire real axis of weakly perturbed linear impulsive differential systems
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