+7 (3412) 91 60 92

## Archive of Issues

Kazakhstan Shymkent; Turkistan
Year
2019
Volume
29
Issue
2
Pages
183-196
 Section Mathematics Title Basis property of a system of eigenfunctions of a second-order differential operator with involution Author(-s) Sarsenbi A.A.a, Turmetov B.Kh.b Affiliations M. Auezov South Kazakhstan State Universitya, Khoja Akhmet Yassawi International Kazakh-Turkish Universityb Abstract In the present paper we study the spectral problem for the second-order differential operators with involution and boundary conditions of Dirichlet type. The Green's function of this boundary problem is constructed. Uniform estimates of the Green's functions for the boundary value problems considered are obtained. The equiconvergence of eigenfunction expansions of two second-order differential operators with involution and boundary conditions of Dirichlet type for any function in $L_{2}(-1,1)$ is established. We use an integral method based on the application of the Green's function of a differential operator with involution and spectral parameter. As a corollary from the equiconvergence theorem, it is proved that the eigenfunctions of the spectral problem form the basis in $L_{2}(-1,1)$ for any continuous complex-valued coefficient $q(x)$. Keywords differential equation with involution, Green's function, eigenfunction expansions, basis UDC 517.927.25 MSC 35K20, 34L10 DOI 10.20537/vm190204 Received 9 February 2019 Language Russian Citation Sarsenbi A.A., Turmetov B.Kh. Basis property of a system of eigenfunctions of a second-order differential operator with involution, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 183-196. References Titchmarsh E.Ch. Eigenfunction expansions associated with second-order differential equations, Oxford: Clarendon Press, 1946. Naimark M.A. Lineinye differentsial'nye operatory (Linear differential operators), Moscow: Nauka, 1969. Coddington E.A., Levinson N. Theory of ordinary differential equations, Malabar, Florida: Krieger publishing company, 1984. Il'in V.A., Kritskov L.V. Properties of spectral expansions corresponding to nonself-adjoint differential operators, Journal of Mathematical Sciences, 2003, vol. 116, issue 5, pp. 3489-3550. https://doi.org/10.1023/A:1024180807502 Abbasova Yu.G., Kurbanov V.M. Convergence of the spectral decomposition of a function from the class $W$$1$ $_{p,m}(G)$, $p>1$, in the vector eigenfunctions of a differential operator of the third order, Ukrainian Mathematical Journal, 2017, vol. 69, issue 6, pp. 839-856. https://doi.org/10.1007/s11253-017-1400-0 Lomov I.S. Estimates of speed of convergence and equiconvergence of spectral decomposition of ordinary differential operators, Izv. Sarat. Univ. (N.S.) Ser. Mat. Mekh. Inform., 2015, vol. 15, issue 4, pp. 405-418 (in Russian). https://doi.org/10.18500/1816-9791-2015-15-4-405-418 Aleroev T.S., Kirane M., Malik S.A. Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition, Electronic Journal of Differential Equations, 2013, vol. 2013, pp. 1-16. https://ejde.math.txstate.edu/Volumes/2013/270/aleroev.pdf Sadybekov M.A., Sarsenbi A.M. Mixed problem for a differential equation with involution under boundary conditions of general form, AIP Conference Proceedings, 2012, vol. 1470, issue 1, pp. 225-227. https://doi.org/10.1063/1.4747681 Kopzhassarova A.A., Lukashov A.L., Sarsenbi A.M. Spectral properties of non-self-adjoint perturbations for a spectral problem with involution, Abstract and Applied Analysis, 2012, vol. 2012, article ID 590781, 5 p. https://doi.org/10.1155/2012/590781 Burlutskaya M.Sh. Mixed problem for a first order partial differential equations with involution and periodic boundary conditions, Computational Mathematics and Mathematical Physics, 2014, vol. 54, issue 1, pp. 1-10. https://doi.org/10.1134/S0965542514010059 Sadybekov M.A. Sarsenbi A.M. Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution, Differential Equations, 2012, vol. 48, no. 8, pp. 1112-1118. https://doi.org/10.1134/S001226611208006X Kopzhassarova A., Sarsenbi A. Basis properties of eigenfunctions of second-order differential operators with involution, Abstract and Applied Analysis, 2012, vol. 2012, article ID 576843, 6 p. https://doi.org/10.1155/2012/576843 Sarsenbi A.M., Tengaeva A.A. On the basis properties of root functions of two generalized eigenvalue problems, Differential Equations, 2012, vol. 48, no. 2, pp. 306-308. https://doi.org/10.1134/S0012266112020152 Sarsenbi A.M. Unconditional bases related to a nonclassical second-order differential operator, Differential Equations, 2010, vol. 46, no. 4, pp. 509-514. https://doi.org/10.1134/S0012266110040051 Kritskov L.V., Sarsenbi A.M. Spectral properties of a nonlocal problem for a second-order differential equation with an involution, Differential Equations, 2015, vol. 51, no. 8, pp. 984-990. https://doi.org/10.1134/S0012266115080029 Kritskov L.V., Sarsenbi A.M. Basicity in $L_p$ of root functions for differential equations with involution, Electronic Journal of Differential Equations, 2015, vol. 2015, no. 278, pp. 1-9. https://ejde.math.txstate.edu/Volumes/2015/278/kristkov.pdf Przeworska-Rolewicz D. Equations with transformed argument. Algebraic approach, Amsterdam, Warsawa: PWN Elsevier, 1973. Wiener J. Generalized solutions of functional differential equations, World Scientific, 1993. https://doi.org/10.1142/1860 Cabada A., Tojo F.A.F. Existence results for a linear equation with reflection, non-constant coefficient and periodic boundary conditions, Journal of Mathematical Analysis and Applications, 2014, vol. 412, no. 1, pp. 529-546. https://doi.org/10.1016/j.jmaa.2013.10.067 Full text