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On a nonlocal boundary value problem of first kind in differential and difference treatments. (Russian) Zbl 0691.34012

The nonlocal boundary-value problem \[ (k(x)u')'+r(x)u'-q(x)u=-f(x)\quad (0<x<1),\quad u(0)=0,\quad u(1)=\alpha u(\zeta)-\beta u(\eta), \] where \(\zeta\),\(\eta\in (0,1)\), \(\alpha\),\(\beta\geq 0\), is considered. The following assumptions are made: \(k\in C^ 3[0,1]\), \(r,q,f\in C^ 2[0,1]\), \(k(x)\geq m_ 0>0\), \(| r(x)| <\mu\), q(x)\(\geq 0\) for all \(x\in [0,1]\), \(\mu <m_ 0\), \(\alpha\)-\(\beta\leq 1\) if \(\zeta\leq \eta\), \(\alpha\leq 1\) if \(\zeta >\eta\). The author proves that the above problem has a unique solution. A corresponding finite difference equation also has a unique solution, and this solution tends to the solution of the boundary-value problem for the differential equation with second order accuracy as the step size tends to zero, with respect to uniform convergence and in the \(W^ 1_ 2\)- and \(W^ 2_ 2\)-norms.
Reviewer: M.Möller

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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