On the necessary and sufficient conditions for the convergence of the difference schemes for the general boundary value problem for the linear systems of ordinary differential equations

summary:We consider the numerical solvability of the general linear boundary value problem for the systems of linear ordinary differential equations. Along with the continuous boundary value problem we consider the sequence of the general discrete boundary value problems, i.e. the corresponding general difference schemes. We establish the effective necessary and sufficient (and effective sufficient) conditions for the convergence of the schemes. Moreover, we consider the stability of the solutions of general discrete linear boundary value problems, in other words, the continuous dependence of solutions on the small perturbation of the initial dates. In the direction, there are obtained the necessary and sufficient condition, as well. The proofs of the results are based on the concept that both the continuous and discrete boundary value problems can be considered as so called generalized ordinary differential equation in the sense of Kurzweil. Thus, our results follow from the corresponding well-posedness results for the linear boundary value problems for generalized differential equations

On boundary value problems for systems of nonlinear generalized ordinary differential equations

summary:A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $${\rm d} x={\rm d} A(t)\cdot f(t,x),\quad h(x)=0$$ is established, where $f\colon [a,b]\times \mathbb {R}^n\to \mathbb {R}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A\colon [a,b]\to \mathbb {R}^{n\times n}$ with bounded total variation components, and $h\colon \operatorname {BV}_s([a,b],\mathbb {R}^n)\to \mathbb {R}^n$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x(t_1(x))=\mathcal {B}(x)\cdot x(t_2(x))+c_0,$ where $t_i\colon \operatorname {BV}_s([a,b],\mathbb {R}^{n})\to [a,b]$ $(i=1,2)$ and $\mathcal {B}\colon \operatorname {BV}_s([a,b],\mathbb {R}^{n})\to \mathbb {R}^n$ are continuous operators, and $c_0\in \mathbb {R}^n$