Weakly perturbed boundary-value problems for systems of integro-differential equations with impulsive action

Abstract

The weakly perturbed $BVP’s$ for impulsive integro-differential systems are
considered. Under the assumption that the generating problem

(for $\varepsilon = 0 $) does not have solutions on the space

$ W^1_2 [a, b]  $  for some nonhomogeneities and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these problems on the space $ D_2([a, b]  \ { {\tau_i\} I } ) $ in the form of a Laurent series in powers of small parameter $\varepsilon $ with finitely many terms with negative powers of $\varepsilon $, and we suggest an algorithm of construction of these solutions.