Weakly nonlinear boundary-value problems for operator equations with pulse influence

Abstract

We consider the problem of finding conditions of solvability and algorithms for construction of solutions of weakly nonlinear boundary-value problems for operator equations (with the Noetherian linear part) with pulse influence at fixed times. The method of investigation is based on passing by methods of the Lyapunov-Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle.The study of the qualitative theory of differential systems with pulse influence started in [1,2] has been further developed and extended in a great number of papers. Thus, an approach to the study of pulse periodic boundaryvalue problems for ordinary differential systems proposed in [1] was successfully developed and applied in the case of weakly nonlinear pulse boundary-value problems (with Noetherian linear part) for ordinary differential systems [3] and for linear boundary-value problems for differential systems with concentrated lag [4]. From the point of view of operator theory, one of the essential peculiarities of these problems is the everywhere solvability [5] of the Cauchy problem for initial differential systems. However, there exists a fairly wide class of boundary-value problems for functional differential equations [6] for which the initial operator equation is not everywhere solvable. Precisely this class of problems is considered in the present paper; our purpose is to obtain criteria for the solvability and formulas for the presentation of solutions of linear boundary-value problems for Noetherian operator equations with pulse influence at fixed times. We find conditions of solvability for weakly nonlinear (with Noetherian linear part) pulse boundary-value problems and construct a convergent iterative algorithm for finding solutions. For this purpose, we use criteria of solvability and formulas obtained in [3] for constructing solutions of linear Noetherian operator equations in Banach and Hilbert spaces. By methods of the Lyapunov-Schmidt type, we pass from a pulse boundary-value problem to an operator system and then apply the convergent iterative procedures based on the fixed-point principle [7,8].Denote an interval -~<a<t<b<+~ by l [o, bl,