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Generalized Solutions of the First Boundary Value Problem for a Differential–Difference Equation in Divergence Form on a Finite Interval

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Abstract

We consider the Dirichlet problem for a second-order differential–difference equation in divergence form with variable coefficients on a finite interval \(Q=(0,d) \). Conditions on the right-hand side of the equation ensuring the smoothness of the generalized solution on the entire interval are studied. It is proved that the generalized solution of the problem belongs to the Sobolev space \(W_2^2(Q) \) if the right-hand side is orthogonal in the space \(L_2(Q) \) to finitely many linearly independent functions.

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REFERENCES

  1. Kamenskii, G.A. and Myshkis, A.D., Statement of boundary value problems for differential equations with deviating arguments in leading terms, Differ. Uravn., 1974, vol. 10, no. 3, pp. 409–418.

    Google Scholar 

  2. Kamenskii, A.G., Boundary value problems for equations with formally symmetric differential–difference operators, Differ. Uravn., 1976, vol. 12, no. 5, pp. 815–824.

    MathSciNet  Google Scholar 

  3. Kamenskii, G.A., Myshkis, A.D., and Skubachevskii, A.L., On smooth solutions of a boundary value problem for a differential–difference equation of the neutral type, Ukr. Mat. Zh., 1985, vol. 37, no. 5, pp. 581–585.

    MathSciNet  MATH  Google Scholar 

  4. Skubachevskii, A.L., Elliptic Functional Differential Equations and Applications, vol. 91 of Operator Theory. Advances and Applications, Basel–Boston–Berlin: Birkhäuser, 1997.

  5. Skubachevskii, A.L. and Ivanov, N.O., On generalized solutions of the second boundary value problem for differential–difference equations with variable coefficients, Sovrem. Mat. Fundam. Napravleniya, 2021, vol. 67, no. 3, pp. 576–595.

    MathSciNet  Google Scholar 

  6. Skubachevskii, A.L. and Ivanov, N.O., Generalized solutions of the second boundary-value problem for differential–difference equations with variable coefficients on intervals of noninteger length, Math. Notes, 2022, vol. 111, no. 6, pp. 913–924.

    Article  MathSciNet  MATH  Google Scholar 

  7. Neverova, D.A. and Skubachevskii, A.L., On the classical and generalized solutions of boundary-value problems for difference-differential equations with variable coefficients, Math. Notes, 2013, vol. 94, no. 5, pp. 653–667.

    Article  MathSciNet  MATH  Google Scholar 

  8. Neverova, D.A., Generalized and classical solutions to the second and third boundary-value problem for differential–difference equations, Funct. Differ. Equat., 2014, vol. 21, pp. 47–65.

    MathSciNet  MATH  Google Scholar 

  9. Liiko, V.V. and Skubachevskii, A.L., Strongly elliptic differential–difference equations with mixed boundary conditions in a cylindrical domain, Sovrem. Mat. Fundam. Napravleniya, 2019, vol. 65, no. 4, pp. 635–654.

    Google Scholar 

  10. Liiko, V.V. and Skubachevskii, A.L., Mixed problems for strongly elliptic differential–difference equations in a cylinder, Math. Notes, 2020, vol. 107, no. 5, pp. 770–790.

    Article  MathSciNet  MATH  Google Scholar 

  11. Krasovskii, N.N., Teoriya upravleniya dvizheniem (Motion Control Theory), Moscow: Nauka, 1968.

    Google Scholar 

  12. Osipov, Yu.S., On stabilization of control systems with delay, Differ. Uravn., 1965, vol. 1, no. 5, pp. 605–618.

    MathSciNet  MATH  Google Scholar 

  13. Kryazhimskii, A.V., Maksimov, V.I., and Osipov, Yu.S., About positional modeling in dynamical systems, Prikl. Mat. Mekh., 1983, vol. 47, no. 6, pp. 883–890.

    MathSciNet  Google Scholar 

  14. Skubachevskii, A.L., On the problem of damping a control system with aftereffect, Dokl. Math., 1994, vol. 49, no. 2, pp. 282–286.

    Article  MathSciNet  MATH  Google Scholar 

  15. Adkhamova, A.Sh. and Skubachevskii, A.L., On one problem of damping a nonstationary control system with aftereffect, Sovrem. Mat. Fundam. Napravleniya, 2019, vol. 65, no. 4, pp. 547–556.

    Google Scholar 

  16. Onanov, G.G. and Skubachevskii, A.L., Nonlocal problems in the mechanics of three-layer shells, Math. Model. Nat. Phenom., 2017, vol. 12, no. 6, pp. 192–207.

    Article  MathSciNet  MATH  Google Scholar 

  17. Onanov, G.G. and Tsvetkov, E.L., On the minimum of the energy functional with respect to functions with deviating argument in a stationary problem of elasticity theory, Russ. J. Math. Phys., 1995, vol. 3, no. 4, pp. 491–500.

    MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation within the framework of the state order project no. FSSF-2023-0016.

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Authors and Affiliations

  1. RUDN University, Moscow, 117198, Russia

    A. L. Skubachevskii & N. O. Ivanov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    A. L. Skubachevskii

Authors
  1. A. L. Skubachevskii
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  2. N. O. Ivanov
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Corresponding authors

Correspondence to A. L. Skubachevskii or N. O. Ivanov.

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Translated by V. Potapchouck

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Skubachevskii, A.L., Ivanov, N.O. Generalized Solutions of the First Boundary Value Problem for a Differential–Difference Equation in Divergence Form on a Finite Interval. Diff Equat 59, 880–892 (2023). https://doi.org/10.1134/S0012266123070029

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  • DOI: https://doi.org/10.1134/S0012266123070029

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