On nonlocal problems for ordinary differential equations and on a nonlocal parabolic transmission problem (Q1818339)

The author considers nonlocal boundary value problems for the operator \[ L(\lambda)u=a(x)u''(x)+(Bu)(x)-\lambda u(x) \] with a linear compact operator \(B\) and some linear functionals appearing in boundary conditions. The coefficient \(a\) is assumed to be piecewise constant with the one discontinuity point at which additional transmission conditions are given. The presented results concerning coerciveness and completeness of root functions are a generalization (on the case of discontinuous coefficient \(a\)) of the corresponding results of \textit{G. M. Gubreev} and \textit{A. I. Kovalenko} [Math. Notes 30, 765-770 (1982); translation from Mat. Zametki 30, 543-552 (1981; Zbl 0481.47029)], \textit{A. A. Shkalikov} [Mosc. Univ. Math. Bull. 37, No. 6, 10-20 (1982); translation from Vestn. Mosk. Univ., Ser. I 1982, No. 6, 12-21 (1982; Zbl 0565.34020)] for the case \(a= \text{const.}\), and \textit{S. Ya. Yakubov} [Completeness of root functions of regular differential operators. Pitman Monographs and Surveys in Pure and Applied Mathematics. 71. New York, NY: John Wiley \& Sons, Inc. (1994; Zbl 0833.34081)] for the continuous coefficient \(a\). The obtained results are applied to a nonlocal parabolic transmission problem.