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Asymptotic Properties of Solutions to Differential Equations of Neutral Type

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We consider an important class of differential-difference equations of neutral type and study asymptotic properties of their solutions. We find necessary and sufficient conditions for exponential stability and represent them in geometric terms as a domain in the space of parameters. We analyze the behavior of the solutions on the boundary of the domain where stability is lost by various reasons. We consider asymptotic properties of the solutions together with the corresponding properties of their derivatives.

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  1. 1

    A. A. Andronov and A. T. Maĭer, “The simplest linear systems with retardation,” Avtom. Telemekh. 7, 95 (1946) [in Russian].

    MathSciNet  MATH  Google Scholar 

  2. 2

    N. V. Azbelev, N. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional-Differential Equations (Nauka, Moscow, 1991; World Federation Publishers Company, Atlanta, GA, 1995).

    MATH  Google Scholar 

  3. 3

    A. S. Balandin, “On asymptotic stability of a class of differential-difference equations,” Vestn. PGTU, Mekh., no. 1, 122 (2009) [in Russian].

  4. 4

    A. S. Balandin, “On stability of an implicit differential equation,” in: Contemp. Methods Appl. Math., Control Theory, and Comput. Technol., 68 (Nauchn. Kniga, Voronezh, 2017) [in Russian].

  5. 5

    A. S. Balandin, “On relationship between the fundamental solution and the Cauchy function for neutral functional differential equations,” Appl. Math. Control Sci., no. 1, 13 (2018) [in Russian].

  6. 6

    A. S. Balandin, “On asymptotic behavior of the fundamental solution and the Cauchy function for neutral differential equations,” Vestn. Tambov Univ., Nat. Tech. Sci. 23, 187 (2018) [in Russian].

    Article  Google Scholar 

  7. 7

    A. S. Balandin and V. V. Malygina, “Exponential stability of linear differential-difference equations of neutral type,” Izv. VUZ Matem., no. 7, 17 (2007) [Russ. Math. 51:7, 15 (2007)].

    MathSciNet  Article  Google Scholar 

  8. 8

    A. S. Balandin and V. V. Malygina, “On stability with derivative of a class of differential equations of neutral type,” Appl. Math. Control Sci., No. 1, 22 (2019) [in Russian].

  9. 9

    R. Bellman and K. L. Cooke, Differential-Difference Equations (Academic Press, New York–London, 1963).

    MATH  Google Scholar 

  10. 10

    W. E. Brumley, “On the asymptotic behavior of solutions of differential-difference equations of neutral type,” J. Differ. Equations 7, 175 (1970).

    MathSciNet  Article  Google Scholar 

  11. 11

    O. Diekmann, Ph. Getto, and Y. Nakata, “On the characteristic equation \(\lambda =\alpha _1+(\alpha _2+\alpha _3\lambda )e^{-\lambda }\) and its use in the context of a cell population model,” J. Math. Biol. 72, 877 (2016).

    MathSciNet  Article  Google Scholar 

  12. 12

    L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory of Differential Equations with Deviating Argument (Nauka, Moscow, 1971; Academic Press, New York–London, 1973).

    MATH  Google Scholar 

  13. 13

    G. M. Fichtenholz, Differential and Integral Calculus, Vol. II, (Nauka, Moscow, 1972) [in Russian].

    Google Scholar 

  14. 14

    P. S. Gromova, “Stability of solutions to nonlinear equations of the neutral type in the asymptotically critical case,” Mat. Zametki 1, 715 (1967) [Math. Notes 1, 472 (1968)].

    MathSciNet  Article  Google Scholar 

  15. 15

    P. S. Gromova and A. M. Zverkin, “The trigonometric series whose sum is a continuous function unbounded on the numerical axis, i.e., solution to the equation with deviating argument,” Differ. Uravn. 4, 1774 (1968) [in Russian].

    MATH  Google Scholar 

  16. 16

    W. Hahn, “Zur Stabilität der Lösungen von linearen Differential-Differenzengleichungen mit konstanten Koeffizienten,” Math. Ann. 131, 151 (1956).

    MathSciNet  Article  Google Scholar 

  17. 17

    J. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York–Heidelberg–Berlin, 1977).

    Book  Google Scholar 

  18. 18

    S. Junca and B. Lombard, “Interaction between periodic elastic waves and two contact nonlinearities,” Math. Models Methods Appl. Sci. 22, no. 4, Art. 1150022 (2012).

  19. 19

    S. Junca and B. Lombard, “Stability of a critical nonlinear neutral delay differential equation,” J. Differ. Equations 256, 2368 (2014).

    MathSciNet  Article  Google Scholar 

  20. 20

    V. B. Kolmanovskiĭ and V. R. Nosov, Stability and Periodic Modes of Adaptable Systems with Aftereffect (Nauka, Moscow, 1981) [in Russian].

    Google Scholar 

  21. 21

    A. N. Kolmogorov and A. N. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1989) [in Russian].

    MATH  Google Scholar 

  22. 22

    V. G. Kurbatov, Linear Differential-Difference Equations (Izd. Voronezh. Gos. Univ., Voronezh, 1990) [in Russian].

    MATH  Google Scholar 

  23. 23

    M. A. Lavrent’ev and B. V. Shabat, Methods for the Theory of Functions of a Complex Variable (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  24. 24

    M. V. Mulyukov, “Stability of two-parameter systems of linear autonomous differential equations with bounded delay,” Izv. Inst. Mat. Inform., Udmurt. Gos. Univ. 51, 79 (2018) [in Russian].

    MathSciNet  Article  Google Scholar 

  25. 25

    Yu. I. Neĭmark, Stability of Linearized Systems (Discrete and Distributed) (LKVVIA, Leningrad, 1949) [in Russian].

    Google Scholar 

  26. 26

    I. A. Ozhiganova, “On the domain of asymptotic stability for first-order differential equations with deviating argument,” Trudy Semin. Teor. Differ. Uravn. Otklon. Argument. 1, 52 (1962) [in Russian].

    MathSciNet  Google Scholar 

  27. 27

    Th. Putelat, J. R. Willis, and J. H. P. Dawes, ‘’Wave-modulated orbits in rate-and-state-friction,” Internat. J. Non-Linear Mech. 47, 258 (2012).

  28. 28

    P. M. Simonov and A. V. Chistyakov, “On exponential stability of linear difference-differential systems,” Izv. VUZ, Mat., no. 6, 34 (1997) [Russ. Math. 41:6, 37 (1997)].

    MATH  Google Scholar 

  29. 29

    V. V. Vlasov, “Spectral problems arising in the theory of differential equations with delay,” Sovrem. Mat., Fundam. Napravl. 1, 69 (2003) [J. Math. Sci., New York 124, 5176 (2004)].

    Article  Google Scholar 

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The work was partially supported by the Ministry of Education and Science of Russia (state contract no. FSNM-2020-0028) and the Russian Foundation for Basic Research (project no. 18-01-00928).

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Correspondence to A. S. Balandin or V. V. Malygina.

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Balandin, A.S., Malygina, V.V. Asymptotic Properties of Solutions to Differential Equations of Neutral Type. Sib. Adv. Math. 31, 79–111 (2021).

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  • differential equation of neutral type
  • stability
  • fundamental solution
  • Cauchy function