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On Generalized Solutions of the First Boundary Value Problem for Differential-Difference Equations with Variable Coefficients

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Abstract

We consider the first boundary value problem for a second-order differential-difference equation with variable coefficients on the interval \((0,d)\). The existence of a generalized solution is proved. We investigated the smoothness of generalized solutions on \((0,d)\) and explicitly obtained linearly independent functions to which the right-hand side of the differential-difference equation must be orthogonal.

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REFERENCES

  1. N. N. Krasovskii, Theory of Control of Motion: Linear Systems (Nauka, Moscow, 1968) [in Russian].

    MATH  Google Scholar 

  2. Yu. S. Osipov, ‘‘On stabilization of control systems with delay,’’ Differ. Uravn. 1, 605–618 (1965).

    MathSciNet  MATH  Google Scholar 

  3. A. L. Skubachevskii, ‘‘On the problem of damping a control system with aftereffect,’’ Dokl. Akad. Nauk 335, 157–160 (1994).

    MathSciNet  MATH  Google Scholar 

  4. A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications (Birkhäuser, Basel, 1997).

    MATH  Google Scholar 

  5. A. V. Bitsadze and A. A. Samarskii, ‘‘On some simple generalizations of linear elliptic boundary problems,’’ Sov. Math. Dokl. 10, 398–400 (1969).

    MATH  Google Scholar 

  6. G. A. Kamenskii and A. D. Myshkis, ‘‘On the formulation of boundary value problems for differential equations with deviating argument and several highest terms,’’ Differ. Uravn. 10, 409–418 (1974).

    MathSciNet  Google Scholar 

  7. A. G. Kamenskii, ‘‘Boundary value problems for equations with formally symmetric differential-difference operators,’’ Differ. Uravn. 12, 815–824 (1976).

    MathSciNet  Google Scholar 

  8. G. A. Kamenskii, A. D. Myshkis, and A. L. Skubachevskii, ‘‘Smooth solutions of a boundary value problem for a differential-difference equation of neutral type,’’ Ukr. Mat. Zh. 37, 581–589 (1985).

    MathSciNet  MATH  Google Scholar 

  9. A. L. Skubachevskii and N. O. Ivanov, ‘‘On generalized solutions of the second boundary-value problem for differential-difference equations with variable coefficients,’’ Sovrem. Mat. Fundam. Napravl. 67, 576–595 (2021).

    MathSciNet  Google Scholar 

  10. A. L. Skubachevskii and N. O. Ivanov, ‘‘Generalized solutions of the second boundary-value problem for differential-difference equations with variable coefficients on intervals of noninteger length,’’ Math. Notes 111, 913–924 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  11. D. A. Neverova and A. L. Skubachevskii, ‘‘On the classical and generalized solutions of boundary-value problems for difference-differential equations with variable coefficients,’’ Math. Notes 94, 653–667 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. L. Skubachevskii and V. V. Liiko, ‘‘On a certain property of a regular difference operator with variable coefficients,’’ Complex Variab. Ellipt. Equat. 64, 852–865 (2019).

    MATH  Google Scholar 

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ACKNOWLEDGMENTS

The author wishes to express gratitude to his scientific supervisor A.L. Skubachevskii for problem statement and paying attention to the work. The author also wishes to express gratitude to the reviewer for a number of suggestions that led to the improvement of the paper.

Funding

This work is supported by the Ministry of Science and Higher Education of the Russian Federation (megagrant agreement no. 075-15-2022-1115).

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia (RUDN University), ul. Miklukho-Maklaya 6, 117198, Moscow, Russia

    N. O. Ivanov

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  1. N. O. Ivanov
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Correspondence to N. O. Ivanov.

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(Submitted by A. B. Muravnik)

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Ivanov, N.O. On Generalized Solutions of the First Boundary Value Problem for Differential-Difference Equations with Variable Coefficients. Lobachevskii J Math 43, 2660–2674 (2022). https://doi.org/10.1134/S1995080222130157

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  • DOI: https://doi.org/10.1134/S1995080222130157

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