Abstract: We study the sign and oscillatory properties of the Green function of discontinuous boundary value problem for a fourth-order equation describing small deformations of two rigidly connected rods with elastic support at the connection point. We obtain criterion for the oscillatory property of the Green function.
Keywords: differential equation on the graph, discontinuous boundary value problem, Green's function, oscillatory properties
For citation: Kulaev R. Ch. Oscillatory properties of the green function of discontinuous boundary value problem for equations of the fourth order. Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol.17, no. 1, pp.47-59.
DOI 10.23671/VNC.2015.1.7292
1. Pokornyj Ju. V., Penkin O. M., Prjadiev V. L., Borovskih A. V.,
Lazarev K. P., Shabrov S. A. Differencial'nye Uravnenija na
Geometricheskih Grafah, Moskva, Fizmatlit, 2007, 272 p. (Russian).
2. Pokornyj Ju. V., Bahtina Zh. I., Zvereva M. B., Shabrov S. A.
Oscilljacionnyj Metod Shturma v Spektral'nyh Zadachah, Moskva,
Fizmatlit, 2009, 192 p. (Russian).
3. Pokornyj Ju. V. O znakoreguljarnyh funkcijah Grina nekotoryh
neklassicheskih zadach, Uspekhi Matematicheskikh Nauk [Russian
Mathematical Surveys], 1981, vol. 36, no. 4, pp. 205-206 (Russian).
4. Borovskih A. V., Pokornyj Ju. V. Chebyshev-Haar systems in the
theory of discontinuous Kellogg kernels. Russian Mathematical
Surveys [Uspekhi Matematicheskikh Nauk, 1994, vol. 49, no. 3, pp.
3-42], 1994, vol. 49, no. 3, pp. 1-42.
5. Borovskih A. V., Lazarev K. P., Pokornyj Ju. V. Oscillatory
Spectral Properties of Discontinuous Boundary Value Problems.
Doklady Mathematics [Doklady Akademii Nauk, 1994, vol. 335, no. 4,
pp. 409-412], 1994, vol. 49, no. 2, pp. 328-333.
6. Borovskih A. V., Lazarev K. P., Pokornyj Ju. V. On Kellogg
Kernels in Discontinuous Problems. Proceedings of the Steklov
Institute of Mathematics [Trudy Mat. Inst. Steklov, 1995, vol. 211,
pp. 102-120], 1995, vol. 211, pp. 92-108.
7. Pokornyj Ju. V., Lazarev K. P. Nekotorye oscilljacionnye teoremy
dlja mnogotochechnyh zadach. Differentsial'nye
Uravneniya [Differential Equations], 1987, vol. 23, no. 4, pp.
658-670 (Russian).
8. Borovskih A. V. Sign Regularity Conditions for Discontinuous
Boundary Value Problems. Mathematical Notes [Mat. Zametki, 2003,
vol. 74, no. 5, pp. 643-655], 2003, vol. 74, no. 5, pp. 607-618.
9. Levin A. Ju., Stepanov G. D. One-Dimensional Boundary-Value
Problems with Operators not Reducing the Number of Changes of Sign.
II. Siberian Mathematical Journal[Sibirskii Matematicheskii
Zhurnal, 1976, vol. 17, no. 4, pp. 813-830], 1976, vol. 17, no. 4,
pp. 612-625.
10. Teptin A. L. On the Oscillation of the Spectrum of a Multipoint
Boundary Value Problem. Russian Mathematics (Izvestiya VUZ. Matematika) [Izv. Vyssh. Uchebn. Zaved. Mat., 1999, no. 4(443), pp.
44-53], 1999, no. 4(443), pp. 42-52.
11. Pokornyj Ju. V. Zeros of the Green's Function for the de la
Vallee-Poussin Problem. Sbornik: Mathematics [Mat. Sb., 2008, vol.
199, no. 6, pp. 105-136], 2008, vol. 199, no. 6, pp. 891-921.
12. Derr V. Ja. K obobshhennoj zadache Valle Pussena,
Differentsial'nye Uravneniya [Differential Equations], 1987, vol.
23, no. 11, pp. 1861-1872 (Russian).
13. Kulaev R. Ch. Criterion for the Positiveness of the Green
Function of a Many-Point Boundary Value Problem for a Fourth-Order
Equation. Differential Equations [Differentsial'nye Uravneniya,
2015, vol. 51, no. 2, pp. 161-173], 2015, vol. 51, no. 2, pp.
163-176.
14. Kulaev R. Ch. Oscillation of the Green Function of a Multipoint
Boundary Value Problem for a Fourth-Order Equation. Differential
Equations [Differentsial'nye Uravneniya], 2015, 2015, vol. 51, no.
4, pp 449-463.
15. Levin A. Ju. Non-Oscillation of Solutions of the Equation
\(x(n)+p_1(t)x(n-1)+\ldots+p_n(t)x=0\). Russian Mathematical Surveys
[Uspekhi Mat. Nauk, 1969, vol. 24, no. 2, pp. 43-96], 1969, vol. 24,
no. 2, pp. 43-99.
16. Derr V. Ja. Disconjugacy of Solutions of Linear Differential
Equations. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika.
Komp'yuternye Nauki [The Bulletin of Udmurt University. Mathematics.
Mechanics. Computer Science], 2009, no. 1, pp. 46-89 (Russian).
17. Stepanov G. D. Effective Criteria for the Strong
Sign-Regularity and the Oscillation Property of the Green's
Functions of Two-Point Boundary-Value Problems. Sbornik: Mathematics
[Mat. Sb., 1997, vol. 188, no. 11, pp. 121-159], 1997, vol. 188, no.
11, pp. 1687-1728.
18. Zavgorodnij M. G., Majorova S. P. Ob odnom uravnenii
matematicheskoj fiziki chetvertogo porjadka na grafe, Issledovanija
po Dif. Uravnenijam i Mat. Modelirovaniju [Studies on Dif. Equations
and Math. Modeling], Vladikavkaz, VSC RAS, 2008, pp. 88-102
(Russian).
19. Kulaev R. Ch. Reduction Method for a Fourth-order Equation on a
Graph. Differential Equations [Differentsial'nye Uravneniya, 2014,
vol. 50, no. 3, pp. 296-308], 2014, vol. 50, no. 3, pp. 292-304.
