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 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.

Weakly nonlinear impulsive boundary value problems for systems of integrodifferential equations

“…Conditions for the existence of solutions of systems of linear integrodifferential equations and boundary-value problems for them were studied in research papers [9,10], and research papers [7,8] were dedicated to nonlinear occasion systems of integrodifferential equations and boundary-value problems for them, for such kind of problems there was developed general theory effective methods of finding solutions. In the given research paper by means of Moore-Penrose pseudoinverse matrices and constructive methods nonlinear systems analysis there were investigated conditions for the existence and there were suggested iterative algorithms of constructing solutions of boundary-value problems for weakly nonlinear system of integrodifferential equations with impulsive action.…”

“…Here, we use the assumptions and notation from [7,10,12]: A.t /; B.t /;˚.t / are, respectively, .m n/; .m n/; .n m/ matrices which components are sought in the space L 2 OEa; bI column vectors of matrice˚.t / are linearly independent at OEa; b; the n 1 vector function f .t / 2 L 2 OEa; bI i is an k i -dimensional column vector of constants, E i ; S i are .k i n/ constant matrices such that rank .E i C S i / D k i .i D 1; 2; :::; p/; i. e., the solution of the system is determined by the unique extension through the point of discontinuity…”