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On the stability of solutions of a multipoint boundary value problem for a system of generalized ordinary differential equations. (English) Zbl 0873.34012
The paper deals with the system of generalized ordinary differential equations (1.1) \(dx(t)= dA_0(t)\cdot f_0(t,x(t))\), and the multipoint boundary conditions (1.2) \(x_i(t_i)= \varphi_i(x)\) \((i=1,\dots,n)\), where \(A_0:[a,b]\to \mathbb{R}^{n\times n}\) is a matrix function with bounded variation components, \(f_0:[a,b]\times \mathbb{R}^n\to\mathbb{R}^n\) is a vector-function satisfying the Carathéodory conditions with respect to \(A_0\), \(\varphi_i\), \(i=1,\dots,n\) are continuous functionals, \(t_1,\dots,t_n\in [a,b]\). Together with (1.1), (1.2), a sequence of approximating problems of the same type (1.1m) \(dx(t)= dA_m(t)\cdot f_m(t,x(t))\), (1.2m) \(x_i(t_{im})= \varphi_{im}(x)\) \((i=1,\dots,n)\), is considered here. The author presents various sufficient conditions which guarantee the solvability of (1.1m), (1.2m), for any sufficiently large \(m\). Moreover, the convergence of the solutions of (1.1m), (1.2m), for \(m\to\infty\), to the solution of (1.1), (1.2) is proved. As special cases of (1.1), (1.2) multipoint boundary value problems for systems of ordinary differential equations and difference equations are studied and the above results are applied to them.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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