#### For citation:

Rykhlov V. S. Expansion in Root Functions of Strongly Irregular Pencil of Differential Operators of the Second Order with Multiple Characteristics. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2016, vol. 16, iss. 2, pp. 165-174. DOI: 10.18500/1816-9791-2016-16-2-165-174, EDN: WCNQJH

# Expansion in Root Functions of Strongly Irregular Pencil of Differential Operators of the Second Order with Multiple Characteristics

We consider the quadratic strongly irregular pencil of ordinary second order differential operators with constant coefﬁcients and with a multiple root of the characteristic equation. The amounts of double expansions in biorthogonal Fourier series in the derived chains of such pencils and a necessary and sufﬁcient condition for convergence of these expansions to the expanded vector-valued function are found. This necessary and sufﬁcient condition is a differential equation relating the components of the expanded vector function. At the same time some conditions of smoothness on the components of the expanded vector-valued function and requirements of the vanishing of its components and some of their derivatives at the ends of the main segment are imposed.

- Naimark M. A. Linear Differential Operators. Pt. I. New York, F. Ungar Publ. Co., 1967, 144 p.; Pt. II. New York, F. Ungar Publ. Co., 1968, 352 p. (Russ. ed.: Naimark M. A. Linear Differential Operators. Moscow, Nauka, 1968, 528 p.).
- Shkalikov A. A. Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. J. Soviet Math., 1986, vol. 33, iss. 6, pp. 1311–1342.
- Gurevich A. P., Khromov A. P. First and second order differential operators with weight functions of variable sign. Math. Notes, 1994, vol. 56, iss. 1, pp. 653–661.
- Khromov A. P. Razlozhenie po sobstvennym funktsiiam odnoi kraevoi zadachi tret’ego poriadka [Expansion in the eigenfunctions of a boundary value problem of the third order]. Issledovaniia po teorii operatorov [Researches on the theory of operators], Ufa, 1988, pp. 182–193 (in Russian).
- Dmitriev O. Iu. Expansion on eigenfunctions the differential operator of n-th order with irregular boundary conditions. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2007, vol. 7, iss. 2, pp. 10–14 (in Russian).
- Rykhlov V. S. Expansion in eigenfunctions of quadratic strongly irregular pencils of differential operators of the second order. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2013, vol. 13, iss. 1, part 1, pp. 21–26 (in Russian).
- Vagabov A. I., Abud A. Kh. Quadruple expandability in Fourier’s series on root elements of a differential pencil with quadruple characteristic. Vestnik Dagest. gos. un-ta [Bull. of the Dagestan State Univ.], 2015, vol. 30, iss. 1, pp. 34–39.
- Hromov A. P. Equiconvergence theorems for integrodifferential and integral operators. Math. USSR Sb., 1982, vol. 42, iss. 3, pp. 331–355.

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