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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.

\[ -u''(x) = \rho^{2}u(-x)\text{ on }L^{2}(-1,1) \]

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.

Reviewer: Mihai Pascu (Bucureşti)

### MSC:

34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |

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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\textit{A. M. Sarsenbi}, Differ. Equ. 46, No. 4, 509--514 (2010; Zbl 1214.34082); translation from Differ. Uravn. 46, No. 4, 506--511 (2010)

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### References:

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