On the Correctness of Nonlinear Boundary Value Problems for Systems of Generalized Ordinary Differential Equations

Abstract

Abstract The concept of a strongly isolated solution of the nonlinear boundary value problem dx(t) = dA(t) · f (t, x(t)), h(x) = 0, is introduced, where A : [a, b] → Rn×n is a matrix-function of bounded variation, f : [a, b] × Rn → Rn is a vector-function belonging to a Carathéodory class, and h is a continuous operator from the space of n-dimensional vector-functions of bounded variation into Rn . It is stated that the problems with strongly isolated solutions are correct. Sufficient conditions for the correctness of these problems are given.