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On stability of a differential equation with aftereffect

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Abstract

We study a linear differential equation with bounded aftereffect and establish conditions for the exponential and uniform stability of its solution in the form of domains in the parameter space. We construct examples that show the exactness of boundaries of stability domains for two classes of functional differential equations with concentrated and distributed delays. Along with classical methods of the functional analysis and function theory, we also use the test equations method.

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References

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

    MATH  Google Scholar 

  2. M. A. Krasnosel’skii, P. P. Zabreiko, I. E. Pustyl’nik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions (Nauka, Moscow, 1966) [in Russian].

    MATH  Google Scholar 

  3. A. N. Tikhonov, “Functional Equations of the Volterra Type and Their Applications to Some Problems of Mathematical Physics,” Byull. Mosk. Gos. Univ. Sekts. A. 1(8), 1–25 (1938).

    MathSciNet  Google Scholar 

  4. V. P. Maksimov, Questions of the General Theory of Functional Differential Equations. Selected Papers (Permsk. Gos. Univ., Prikamsk. Soc. Inst., Prikamsk. Sovr. Soc.-Gum. Koll., Perm, 2003) [in Russian].

    Google Scholar 

  5. V. P. Maksimov and L. F. Rakhmatullina, “On Representation of Solutions to Linear Functional Differential Equations,” Differents. Uravn. 9(6), 1026–1036 (1973).

    MATH  Google Scholar 

  6. V. P. Maksimov, “On Cauchy Formula for Functional Differential Equations,” Differents.Uravn. 13(4), 601–606 (1977).

    MATH  MathSciNet  Google Scholar 

  7. N. V. Azbelev, L. M. Berezanskii, P. M. Simonov, and A. V. Chistyakov, “Stability of Differential Systems with Aftereffect. IV,” Differents. Uravn. 29(2), 196–204 (1993).

    MathSciNet  Google Scholar 

  8. Yu. A. Abramovich, “On a Space of Operators Acting between Banach Lattices,” Zap. Nauchn. Semin. LOMI 73, 188–192 (1977).

    MATH  MathSciNet  Google Scholar 

  9. A. V. Bukhvalov, V. B. Korotkov, and A. G. Kusraev, Vector Lattices and Integral Operators (Nauka, Sib. Otd., Novosibirsk, 1992) [in Russian].

    Google Scholar 

  10. L. Kantorovitch and B. Vulich, “Sur la représentation des opérations linéaires,” Compositio Mathematica 5, 119–165 (1938), http://www.numdam.org/item?id=CM_1938__5_ _119_0.

    MathSciNet  Google Scholar 

  11. J. Hale, Theory of Functional Differential Equations (Springer Verlag, New York, 1977; Mir, Moscow, 1984).

    Book  MATH  Google Scholar 

  12. N. V. Azbelev and T. S. Sulavko, “On the Question of the Stability of the Solutions of Differential Equations with Retarded Argument,” Differents. Uravn. 10(12), 2091–2100 (1974).

    MathSciNet  Google Scholar 

  13. N. V. Azbelev and P. M. Simonov, Stability of Solutions to Equations with Ordinary Derivatives (Permsk. Univ., Perm, 2001) [in Russian].

    Google Scholar 

  14. T. L. Sabatulina, “Positiveness Conditions for the Cauchy Function for Differential Equations with Distributed Delay,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, 50–62 (2010) [Russian Mathematics (Iz. VUZ) 54 (11), 44–55 (2010)].

    Google Scholar 

  15. L. Berezansky and E. Braverman, “Linearized Oscillation Theory for Nonlinear Equation with a Distributed Delay,” Appl.Math. and Comp. Model. 48, 287–304 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  16. A. D. Myshkis, Linear Differential Equations with Retarded Argument (Nauka, Moscow, 1972) [in Russian].

    MATH  Google Scholar 

  17. V. V. Malygina, “On Stability of Solutions to Certain Linear Differential Equations with Aftereffect,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 72–85 (1993) [Russian Mathematics (Iz. VUZ) 37 (5), 63–75 (1993)].

    Google Scholar 

  18. V. V. Malygina and K. M. Chudinov, “Stability of Solutions to Differential Equations with Several Variable Delays. I,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 6, 25–36 (2013) [Russian Mathematics (Iz. VUZ) 57 (6), 21–31 (2013)].

    Google Scholar 

  19. V. V. Malygina, “Certain Tests of Stability for Functional-Differential Equations, Resolved with Respect to Derivative,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 7, 46–53 (1992) [RussianMathematics (Iz. VUZ) 36 (7), 44–51 (1992)].

    Google Scholar 

  20. T. Yoneyama, “On the 3/2 Stability Theorem for One-Dimensional Delay-Differential Equations with Unbounded Delay,” J. Math. Anal. Appl. 165, 133–143 (1992).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Perm National Research Polytechnic University, Komsomol’skii pr. 29, Perm, 614990, Russia

    T. L. Sabatulina & V. V. Malygina

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  1. T. L. Sabatulina
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  2. V. V. Malygina
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Correspondence to T. L. Sabatulina.

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Dedicated to Professor N. V. Azbelev on the occasion of his 90th birthday

Original Russian Text © T.L. Sabatulina and V.V. Malygina, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 4, pp. 25–41.

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Sabatulina, T.L., Malygina, V.V. On stability of a differential equation with aftereffect. Russ Math. 58, 20–34 (2014). https://doi.org/10.3103/S1066369X14040045

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