Skip to main content

Exponent Estimation for Stable Solutions of a Certain Class of Differential-Difference Equations


For a differential-difference equation with a positive fundamental solution we obtain exponential stability conditions with exact estimates of the exponent and the coefficient of the exponential decay. These estimates are expressed in terms of the largest of two possible real roots of the characteristic function. We prove that one can obtain exact estimates for any solution by estimating the fundamental solution, taking into account the norm of the initial function. We establish two-sided estimates for the fundamental solution in the case, when equation parameters are given as intervals.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. Pertsev N.V. "Application of M-matrices in construction of exponential estimates for solutions to the Cauchy problem for systems of linear difference and differential equations", Matem. Tr. 16 (2), 111-141 (2013).

    MATH  Google Scholar 

  2. Demidenko G.V., Matveeva I.I. "Asymptotic properties of solutions to delay differential equations", Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Informatika 5 (3), 20-28 (2005).

    MATH  Google Scholar 

  3. Liz E., Pituk M. "Exponential stability in a scalar functional-differential equation", J. Inequal. Appl. 2006, article 37195 (2006).

    MathSciNet  Article  Google Scholar 

  4. Azbelev N.V., Maksimov V.P., Rakhmatullina L.F. Introduction to the Theory of Linear Functional Differential Equations (Nauka, Moscow, 1991) [in Russian].

    MATH  Google Scholar 

  5. Myshkis A.D. Linear Differential Equations with Retarded Arguments (Nauka, Moscow, 1972).

    Google Scholar 

  6. Ryabov Yu.A. "Some asymptotic properties of linear systems with a small time lag", Dokl. Akad. Nauk SSSR 151 (1), 52-54 (1963).

    MathSciNet  Google Scholar 

  7. Azbelev N.V., Simonov P.M. Stability of Solutions of Ordinary Differential Equations (Izd-vo Permsk. Univ., Perm, 2001).

    Google Scholar 

  8. Agarwal R.P., Berezansky L., Braverman E., Domoshnitsky A. Nonoscillation Theory of Functional Differential Equations with Applications (Springer, New York, 2012).

    Book  Google Scholar 

  9. Chudinov K.M. "Asymptotic properties of positive solutions to autonomous differential equations with aftereffect", in: Sovremennye Metody Prikl. Matem., Teorii Upravleniya i Komp'uternykh Tekhn. (PMTUKT-2015): Sb. Tr. Konf., Voronezh, September 21–26, 2015, 386-388 (Nauchn. Kniga, Voronezh, 2015) [in Russian].

    Google Scholar 

  10. Sabatulina T., Malygina V. "On positiveness of the fundamental solution for a linear autonomous differential equation with distributed delay", Electron. J. Qual. Theory Differ. Equat. 61, 1-16 (2014).

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. El'sgol'tz L.E., Norkin S.B. Introduction to the Theory of Differential Equations with Deviating Arguments (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  13. Hale J.K. Theory of Functional Differential Equations (Mir, Moscow, 1984) [in Russian].

    MATH  Google Scholar 

  14. Malygina V.V., Chudinov K.M. "Stability of solutions to differential equations with several variable delays. II", Russian Math. (Iz. VUZ) 57 (7), 1-12 (2013).

    MathSciNet  Article  Google Scholar 

  15. Sabatulina T.L. "Oscillating and sign-definite solutions to autonomous functional-differential equations", J. Math. Sci. 230 (5), 766-769 (2018).

    MathSciNet  Article  Google Scholar 

  16. Györi I., Ladas G. Oscillation Theory of Delay Differential Equations: with Applications (Clarendon Press, Oxford, 1991).

    MATH  Google Scholar 

Download references


This work was supported by the Ministry of Education and Science of the Russian Federation, state assignment FSNM-2020-0028.

Author information

Authors and Affiliations


Corresponding author

Correspondence to V. V. Malygina.

Additional information

Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 12, pp. 67–79.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Malygina, V.V. Exponent Estimation for Stable Solutions of a Certain Class of Differential-Difference Equations. Russ Math. 65, 56–67 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • functional differential equation
  • fundamental solution
  • exponential stability
  • exponent estimate