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Unconditional bases related to a nonclassical second-order differential operator. (English. Russian original) Zbl 1214.34082

Differ. Equ. 46, No. 4, 509-514 (2010); translation from Differ. Uravn. 46, No. 4, 506-511 (2010).
One introduces the notion of regular boundary conditions for the second order differential equation with deviating argument
\[ -u''(x) = \rho^{2}u(-x)\text{ on }L^{2}(-1,1) \]
and a corresponding system of root functions is defined. One proves that the system of root functions is an unconditional basis in \(L^{2}(-1,1)\). The main idea of proof is to reduce the problem to the case of a fourth order ordinary differential equation with strongly regular (in Birkhoff’s sense) boundary conditions.

MSC:

34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34K10 Boundary value problems for functional-differential equations
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