Numerical-analytic method for the investigation of boundary-value problems for semilinear systems of differential equations

Abstract

We propose a new numerical-analytic algorithm for the investigation of boundary-value problems for semilinear systems of differential equations.One of branches of the theory of ordinary differential equations that have been rapidly developed in recent years is the theory of boundary-value problems. There are many works devoted to the investigation of different aspects of this theory (see, e.g., [1-9]), among which an important place is occupied by problems of the existence and approximate construction of solutions of linear and nonlinear boundary-value problems.We consider the problem of the existence and approximate construction of solutions of systems of differential equations with separated linear partwith linear boundary conditions`x D˛;˛2 R n ;where A.t / is an n n matrix with real components, A.t/ 2 C OEa; b;`is a linear vector functional,`W C OEa; b ! R n ; and˛2 R n is a constant vector. According to the Riesz theorem [10], for any linear functional`defined on the space of functions continuous on OEa; b; there exists a matrix-valued function C.t/ of bounded variation such that this linear functional can be represented in the form of a Riemann-Stieltjes integral. Therefore, the boundary conditions (2) can be rewritten in the form b Z a dC.t/ x.t/ D˛: Remark 1. Every function C.t / of bounded variation defined on OEa; b can be represented in the formwhere C 1 .t / 2 C OEa; b and C 2 .t/ is a constant function on OEa; b except for a finite or countable set of points at which C 2 .t / has discontinuities of the first kind. In the case of a finite set of points of discontinuity, problem (1), (2) becomes a linear multipoint boundary-value problem. In this case, we have C 1 .t / D 0; and the points of discontinuity of the function C 2 .t/ are the points at which the solution takes the values appearing in the boundary conditions. We assume that the following condition is satisfied: