Nonlocal boundary-value problem of the second kind for a Sturm-Liouville operator. (English. Russian original) Zbl 0668.34024

Differ. Equations 23, No. 8, 979-987 (1987); translation from Differ. Uravn. 23, No. 8, 1422-1431 (1987).
Consider the differential and difference treatment of the nonlocal boundary-value problem of the second kind \[ (1)\quad (k(x)u')'-q(x)u=- f(x),\quad x\in (0,1),\quad u(0)=0,\quad \Pi (1)=\sum^{m}_{\ell =1}\alpha_{\lambda}\Pi (\xi_{\ell}) \] where \(\Pi (x)=k(x)u'(x)\) is a flow; \(0\leq \xi_ 1<\xi_ 2<\dots<\xi_ m<1\) and the constants \(\alpha_ 1,\alpha_ 2,...,\alpha_ m\) are either positive or negative. A. V. Bitsadze and A. A. Somarskij [Dokl. Akad. Nauk SSSR 185, No.4, 739-740 (1969; Zbl 0187.35501)] studied the problem. In this paper the authors prove that the difference problem has a unique solution tending (as the step\(\to 0)\) to the solution of the differential problem with second-order accuracy with respect to the step, both in the uniform metric and in the \(w^ 1_ 2\) and \(w^ 2_ 2\)-metrics.
Reviewer: J.H.Tian


34L99 Ordinary differential operators


Zbl 0187.35501