Weakly Nonlinear Systems of Integrodifferential Equations

Abstract

517.929We establish necessary and sufficient conditions for the existence of the solution of a weakly nonlinear system of integrodifferential equations. We propose a convergent iterative procedure for finding solutions of the system and establish a relationship between the necessary and sufficient conditions. Statement of the Problem and Auxiliary ResultsBy using the theory of pseudoinverse Moore-Penrose matrices [1-3], we investigate the conditions of solvability and propose an iterative algorithm for the construction of solutions of weakly nonlinear systems of integrodifferential equations with small parameter "We seek conditions for the existence of a solutionIn what follows, this solution is called a generating solution of the nonlinear system (1). Here, A.t / and B.t / are .m ⇥ n/ matrices,ˆ.t / is an .n ⇥ m/ matrix, f .t / is an .n ⇥ 1/ matrix, and K.t; s/ is an .n ⇥ n/ matrix whose components belong to the space L 2 OEa; bç; the columns of the matrixˆ.t / are linearly independent of OEa; bç; and Z.x.t; "/; t; "/ is an n -dimensional vector function nonlinear with respect to the first component and such that Z.�; t; "/ 2 C 1OEkx � x 0 k  qç; Z.x.�; "/; �; "/ 2 L 2 OEa; bç; Z.x.t; �/; t; �/ 2 C OE0; " 0 ç:First, we present the well-known criterion of solvability of the generating linear system (2) of integrodifferential equations [1]. Theorem 1. System (2) is solvable in the space D 2 OEa; bç of absolutely continuous vector functions on OEa; bç whose derivative belongs to the space L 2 OEa; bç if and only if the inhomogeneity f .t / 2 L 2 OEa; bç satisfies the condition

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.…”