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Proof of the completeness of the system of eigenfunctions for one boundary-value problem for the fractional differential equation

  • Received: 21 April 2019 Accepted: 12 June 2019 Published: 20 June 2019
  • MSC : 26A33, 34A08

  • The present paper is devoted to the spectral analysis of operators induced by differential expressions of fractional order and boundary conditions of Sturm-Liouville type. In particular, this paper establishes the completeness of the system of eigenfunctions and associated functions of one class for non-self-adjoint integral operators associated with boundary-value problems for fractional-order differential equations.

    Citation: Mukhamed Aleroev, Hedi Aleroeva, Temirkhan Aleroev. Proof of the completeness of the system of eigenfunctions for one boundary-value problem for the fractional differential equation[J]. AIMS Mathematics, 2019, 4(3): 714-720. doi: 10.3934/math.2019.3.714

    Related Papers:

  • The present paper is devoted to the spectral analysis of operators induced by differential expressions of fractional order and boundary conditions of Sturm-Liouville type. In particular, this paper establishes the completeness of the system of eigenfunctions and associated functions of one class for non-self-adjoint integral operators associated with boundary-value problems for fractional-order differential equations.


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    [3] T. S. Aleroev, Boundary-Value Problems for Differential Equations with Fractional Derivatives, Doctoral Degree Thesis, University Moscow State University of Civil Engineering, Moscow, 2000.
    [4] T. S. Aleroev, H. T. Aleroeva, N. M. Nie, et al. Boundary value problems for differential equations of fractional order, Mem. Diff. Equ. Math. Phys., 49 (2010), 19-82.
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    [9] A. V. Agibalova, On the completeness of the systems of root functions of a fractional-order differential operator with matrix coefficients, Mat. Zametki, 88 (2010), 317-320. doi: 10.4213/mzm8806
    [10] M. M. Malamud, Similarity of Volterra operators and related problems in the theory of differential equations of fractional orders (Russian), translation in Trans. Moscow Math. Soc., 55 (1994), 57-122.
    [11] M. M. Malamud and L. L. Oridoroga, Analog of the Birkhoff theorem and the completeness results for fractional order differential equations, Russ. J. Math. Phys., 8 (2001), 287-308.
    [12] M. M. Malamud and L. L. Oridoroga, On some questions of the spectral theory of ordinary differential fractional-order equation, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 9 (1998), 39-47.
    [13] M. M. Malamud, Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations, In: Kochubei, A., Luchko, Y. Editors, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, Berlin, Boston: Walter de Gruyter GmbH, 2019.
    [14] T. S. Aleroev, On one class of operators associated with differential equations of fractional order, Sib. Mat. Zh., 46 (2005), 963-968. doi: 10.1007/s11202-005-0093-z
    [15] T. S. Aleroev and H. T. Aleroeva, Problems of Sturm-Liouville type for differential equations with fractional derivatives, In: Kochubei, A., Luchko Y. Editors, Handbook of Fractional Calculus with Applications. Volume 4: Fractional Differential Equations, Berlin, Boston: De Gruyter, 2019.
    [16] M. M. Dzhrbashian, The boundary-value problem for a differential fractional-order operator of the Sturm-Liouville type, Izv. Akad. Nauk ArmSSR, Ser. Mat., 5 (1970), 71-96.
    [17] T. S. Aleroev, Boundary value problems for differential equations of fractional order, Sib. Electr. Mat.Izv., 10 (2013), 41-55.
    [18] P. Ma, Y. Li and J. Zhang, Symmetry and nonexistence of positive solutions for fractional systems, Commun. Pure Appl. Anal., 17 (2018), 1053-1070. doi: 10.3934/cpaa.2018051
    [19] P. Ma and J. Zhang, Existence and multiplicity of solutions for fractional Choquard equations, Nonlinear Anal., 164 (2017), 100-117. doi: 10.1016/j.na.2017.07.011
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