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On exponential stability of second order delay differential equations

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Abstract

We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays.

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References

  1. N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina: Introduction to the Theory of Functional-Differential Equations. Nauka, Moskva, 1991. (In Russian. English summary.)

    MATH  Google Scholar 

  2. L. Berezansky, E. Braverman, A. Domoshnitsky: Stability of the second order delay differential equations with a damping term. Differ. Equ. Dyn. Syst. 16 (2008), 185–205.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. A. Burton: Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, Mineola, 2006.

    MATH  Google Scholar 

  4. T. A. Burton: Stability and Periodic Solutions of Ordinary and Functional-Differential Equations. Mathematics in Science and Engineering 178, Academic Press, Orlando, 1985.

    MATH  Google Scholar 

  5. T. A. Burton, T. Furumochi: Asymptotic behavior of solutions of functional differential equations by fixed point theorems. Dyn. Syst. Appl. 11 (2002), 499–519.

    MATH  MathSciNet  Google Scholar 

  6. T. A. Burton, L. Hatvani: Asymptotic stability of second order ordinary, functional, and partial differential equations. J. Math. Anal. Appl. 176 (1993), 261–281.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Cahlon, D. Schmidt: Stability criteria for certain second-order delay differential equations with mixed coefficients. J. Comput. Appl. Math. 170 (2004), 79–102.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Cahlon, D. Schmidt: Stability criteria for certain second order delay differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 10 (2003), 593–621.

    MATH  MathSciNet  Google Scholar 

  9. A. Domoshnitsky: Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term. J. Inequal. Appl. (2014), 2014:361, 26 pages.

  10. A. Domoshnitsky: Unboundedness of solutions and instability of differential equations of the second order with delayed argument. Differ. Integral Equ. 14 (2001), 559–576.

    MATH  MathSciNet  Google Scholar 

  11. A. Domoshnitsky: Componentwise applicability of Chaplygins theorem to a system of linear differential equations with time-lag. Differ. Equations 26 (1990), 1254–1259; translation from Differ. Uravn. 26 (1990), 1699–1705. (In Russian.)

    MathSciNet  Google Scholar 

  12. Z. Došlá, I. Kiguradze: On boundedness and stability of solutions of second order linear differential equations with advanced arguments. Adv. Math. Sci. Appl. 9 (1999), 1–24.

    MATH  MathSciNet  Google Scholar 

  13. L. H. Erbe, Q. Kong, B. G. Zhang: Oscillation Theory for Functional Differential Equations. Pure and Applied Mathematics 190, Marcel Dekker, New York, 1995.

    Google Scholar 

  14. T. Erneux: Applied Delay Differential Equations. Surveys and Tutorials in the Applied Mathematical Sciences 3, Springer, New York, 2009.

    MATH  Google Scholar 

  15. V. N. Fomin, A. L. Fradkov, V. A. Yakubovich: Adaptive Control of Dynamical Objects. Nauka, Moskva, 1981. (In Russian.)

    MATH  Google Scholar 

  16. D. V. Izyumova: On the boundedness and stability of the solutions of nonlinear second order functional-differential equations. Soobshch. Akad. Nauk Gruz. SSR 100 (1980), 285–288. (In Russian.)

    MATH  MathSciNet  Google Scholar 

  17. V. Kolmanovskii, A. Myshkis: Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and Its Applications 463, Kluwer Academic Publishers, Dordrecht, 1999.

    Book  MATH  Google Scholar 

  18. G. S. Ladde, V. Lakshmikantham, B. G. Zhang: Oscillation Theory of Differential Equations with Deviating Arguments. Pure and Applied Mathematics 110, Marcel Dekker, New York, 1987.

    Google Scholar 

  19. N. Minorsky: Nonlinear Oscillations. D. Van Nostrand Company, Princeton, 1962.

    MATH  Google Scholar 

  20. A. D. Myshkis: Linear Differential Equations with Retarded Argument. Izdat. Nauka, Moskva, 1972. (In Russian.)

    MATH  Google Scholar 

  21. M. Pinto: Asymptotic solutions for second order delay differential equations. Nonlinear Anal., Theory Methods Appl. 28 (1997), 1729–1740.

    Article  MATH  Google Scholar 

  22. L. S. Pontryagin: On the zeros of some elementary transcendental functions. Am. Math. Soc., Transl., II. Ser. 1 (1955), 95–110; Izv. Akad. Nauk SSSR, Ser. Mat. 6 (1942), 115–134. (In Russian.)

    MathSciNet  Google Scholar 

  23. B. Zhang: On the retarded Liénard equation. Proc. Am. Math. Soc. 115 (1992), 779–785.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Texas A&M University, 401 Joe Routt Blvd, Kingsville, Texas, TX 78363-8202, USA

    Ravi P. Agarwal

  2. Department of Mathematics and Computer Science, Ariel University, Qiryat University, 40700, Ariel, Israel

    Alexander Domoshnitsky & Abraham Maghakyan

Authors
  1. Ravi P. Agarwal
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  2. Alexander Domoshnitsky
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  3. Abraham Maghakyan
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Correspondence to Ravi P. Agarwal.

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Agarwal, R.P., Domoshnitsky, A. & Maghakyan, A. On exponential stability of second order delay differential equations. Czech Math J 65, 1047–1068 (2015). https://doi.org/10.1007/s10587-015-0227-9

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  • DOI: https://doi.org/10.1007/s10587-015-0227-9

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