Necessary and sufficient conditions for the existence of solutions of weakly nonlinear boundary-value problem of systems of integrodifferential equations with impulsive action at fixed points of time are obtained. Convergent iterative procedure of the solution and establish the connection between the necessary condition and the sufficient condition are presented.
The weakly perturbed BVP's for impulsive integro-differential systems are considered. Under the assumption that the generating problem (for ε = 0) does not have solutions on the space W12[a,b] for some inhomogeneity and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these problems on the space D2([a,b]{τi}I) 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.
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