Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

The weakly perturbed linear nonhomogeneous impulsive systems in the form ẋ=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