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Zbl 1214.34082
Sarsenbi, A.M.
Unconditional bases related to a nonclassical second-order differential operator. (English. Russian original)
[J] Differ. Equ. 46, No. 4, 509-514 (2010); translation from Differ. Uravn. 46, No. 4, 506-511 (2010). ISSN 0012-2661; ISSN 1608-3083/e

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.
[Mihai Pascu (Bucureşti)]

MSC 2000:: *34L10 Eigenfunction expansions, etc. (ODE)
34K10 Boundary value problems for functional-differential equations

Keywords: differential equations with deviating argument; regular boundary conditions; root functions; unconditional bases

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