Nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in the differential and difference treatments. (Russian) Zbl 0636.34019

Consider the nonlocal boundary value problem of the first type \([k(x)u']'-q(x)u=-f(x),\) \(0<x<1\) \(u(0)=0\), \(u(1)=\sum^{m}_{1}\alpha_{\ell}u(\xi_{\ell})\) where \(0<\xi_ 1<\xi_ 2<...<\xi_ m<1\) and \(\alpha_{\ell}\) are either all of them nonpositive or all of them nonnegative; also \(-\infty <\sum^{m}_{1}\alpha_{\ell}\leq 1\). The following assumptions are true: i) \(k\in C^{(3)}[0,1]\); ii) \(q,f\in C^{(2)}[0,1]\); iii) \(k(x)\geq m_ 0>0\); q(x)\(\geq 0\) on [0,1]. Then there exists a unique solution of the boundary problem which belongs to \(C^{(4)}[0,1]\). Also a difference scheme can be shown such that the solution of the difference problem exists and tends to the solution of the boundary value problem when the step \(h\to 0\).
Reviewer: V.Rǎsvan


34L99 Ordinary differential operators
65L10 Numerical solution of boundary value problems involving ordinary differential equations