Malkhaz Ashordia

On The Stability of Solutions of The Multipoint Boundary Value Problem for The System of Generalized Ordinary Differential Equations

abstract:
The boundary value problem \begin{equation} dx(t)=dA(t)\cdot f\big(t,x(t)\big),\;\;\;x_i(t_i)=\vf_i(x)\;\;\;(i=1,\dots,n)\tag{$1$} \end{equation} is considered, where $A:[a,b]\to R^{n\times n}$ is a matrix-function with of components bounded variation, $f$ is a vector-function belonging to the Caratheodory class corresponding to $A$; $t_1,\dots,t_n\in[a,b]$, $x=(x_i)_{i=1}^n$ and $\vf_1,\dots,\vf_n$ are the continuous functionals, in general nonlinear, given on the space of all vector-functions of bounded variation. The sequence of problems \begin{gather} dx(t)=dA_m(t)\cdot f_m\big(t,x(t)\big),\;\;\;x_i(t_{im})=\vf_{im}(x)\;\;\; (i=1,\dots,n)\tag{$1_m$}\\ (m=1,2,\ldots)\notag \end{gather} is considered along with (1). Sufficient conditions are given which guarantee both solvability of the problem $(1_m)$ for any sufficiently large $m$ and convergence of its solutions as $m\to +\infty$ to the solution of the problem (1), provided this problem is solvable. Difference schemes of numerical solutions for the multipoint differential and difference problems are constructed.

Mathematics Subject Classification: 34B10.

Key words and phrases: Multipoint boundary value problem, system of generalized differential equations, correct problem, system of ordinary differential and difference equations.