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Hyers–Ulam Stability for a Class of Perturbed Hill’s Equations

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Abstract

In this note we formulate sufficient conditions under which a certain class of nonlinear and nonautonomous differential equations of second order is Hyers–Ulam stable. This class consists of equations obtained by perturbing Hill’s equation of the form \(x''=(\lambda ^2(t)-\lambda '(t))x\).

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References

  1. Backes, L., Dragičević, D.: Shadowing for nonautonomous dynamics. Adv. Nonlinear Stud. 19, 425–436 (2019)

    Article  MathSciNet  Google Scholar 

  2. Backes, L., Dragičević, D.: Shadowing for infinite dimensional dynamics and exponential trichotomies. Proc. R. Soc. Edinb. Sect. A 151, 863–884 (2021)

    Article  MathSciNet  Google Scholar 

  3. Barbu, D., Buşe, C., Tabassum, A.: Hyers–Ulam stability and discrete dichotomy. J. Math. Anal. Appl. 42, 1738–1752 (2015)

    Article  MathSciNet  Google Scholar 

  4. Barbu, D., Buşe, C., Tabassum, A.: Hyers–Ulam stability and exponential dichotomy of linear differential periodic systems are equivalent. Electron. J. Qual. Theory Differ. Equ. 2015(58), 1–12 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Brzdek, J., Popa, D., Raşa, I., Xu, B.: Ulam Stability of Operators. Academic Press, London (2018)

    MATH  Google Scholar 

  6. Buşe, C., O’Regan, D., Saierli, O., Tabassum, A.: Hyers–Ulam stability and discrete dichotomy for difference periodic systems. Bull. Sci. Math. 140, 908–934 (2016)

    Article  MathSciNet  Google Scholar 

  7. Buşe, C., O’Regan, D., Saierli, O.: Hyers–Ulam stability for linear differences with time dependent and periodic coefficients: the case when the monodromy matrix has simple eigenvalues. Symmetry 2019(11), 339 (2019)

    Article  Google Scholar 

  8. Buşe, C., Lupulescu, V., O’Regan, D.: Hyers–Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients. Proc. R. Soc. Edinb. Sect. A 150, 2175–2188 (2020)

    Article  MathSciNet  Google Scholar 

  9. Chicone, C.: Ordinary Differential Equations with Applications. Springer, Berlin (2006)

    MATH  Google Scholar 

  10. Coppel, W.A.: Dichotomies in Stability Theory. Springer, Berlin (1978)

    Book  Google Scholar 

  11. Dragičević, D.: On the Hyers–Ulam stability of certain nonautonomous and nonlinear difference equations. Aequat. Math. (2021). https://doi.org/10.1007/s00010-020-00774-7

    Article  MATH  Google Scholar 

  12. Dragičević, D.: Hyers–Ulam stability for nonautonomous semilinear dynamics on bounded interval. Mediterr. J. Math. 18, Paper No. 71, 9 pp (2021)

  13. Dragičević, D., Pituk, M.: Shadowing for nonautonomous difference equations with infinite delay. Appl. Math. Lett. 120, 107284 (2021)

    Article  MathSciNet  Google Scholar 

  14. Fukutaka, R., Onitsuka, M.: A necessary and sufficient condition for Hyers–Ulam stability of first-order periodic linear differential equations. Appl. Math. Lett. 100, 106040 (2020)

    Article  MathSciNet  Google Scholar 

  15. Fukutaka, R., Onitsuka, M.: Best constant in Hyers–Ulam stability of first-order homogeneous linear differential equations with a periodic coefficient. J. Math. Anal. Appl. 473, 1432–1446 (2019)

    Article  MathSciNet  Google Scholar 

  16. Fukutaka, R., Onitsuka, M.: Ulam stability for a class of Hill’s equations. Symmetry 11, 1483 (2019)

    Article  Google Scholar 

  17. Fukutaka, R., Onitsuka, M.: Best constant for Ulam stability of Hill’s equations. Bull. Sci. Math. 163, 102888 (2020)

    Article  MathSciNet  Google Scholar 

  18. Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, 1135–1140 (2004)

    Article  MathSciNet  Google Scholar 

  19. Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order II. Appl. Math. Lett. 19, 854–858 (2006)

    Article  MathSciNet  Google Scholar 

  20. Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order III. J. Math. Anal. Appl. 311, 139–146 (2005)

    Article  MathSciNet  Google Scholar 

  21. Li, T., Zada, A.: Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces. Adv. Differ. Equ. 153, 1–8 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Popa, D., Raşa, I.: On the Hyers–Ulam stability of the linear differential equations. J. Math. Anal. Appl. 381, 530–537 (2011)

    Article  MathSciNet  Google Scholar 

  23. Popa, D., Raşa, I.: Hyers–Ulam stability of the linear differential operator with nonconstant coefficients. Appl. Math. Comput. 219, 1562–1568 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Wang, J., Fečkan, M., Tian, Y.: Stability analysis for a general class of non-instantaneous impulsive differential equations. Mediterr. J. Math. 14, 46 (2017)

    Article  MathSciNet  Google Scholar 

  25. Wang, J., Fečkan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)

    Article  MathSciNet  Google Scholar 

  26. Zada, A., Zada, B.: Hyers–Ulam stability and exponential dichotomy of discrete semigroup. Appl. Math. E-Notes 19, 527–534 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Zada, A., Shah, S.O., Shah, R.: Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problem. Appl. Math. Comput. 271, 512–518 (2015)

    MathSciNet  MATH  Google Scholar 

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Funding

D.D. was supported in part by Croatian Science Foundation under the project IP-2019-04-1239 and by the University of Rijeka under the projects uniri-prirod-18-9 and uniri-prprirod-19-16.

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  1. Department of Mathematics, University of Rijeka, Rijeka, Croatia

    Davor Dragičević

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  1. Davor Dragičević
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Correspondence to Davor Dragičević.

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Dragičević, D. Hyers–Ulam Stability for a Class of Perturbed Hill’s Equations. Results Math 76, 129 (2021). https://doi.org/10.1007/s00025-021-01442-1

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