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Ulam Type Stabilities of n-th Order Linear Differential Equations Using Gronwall’s Inequality

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Abstract

In this paper, the stabilities in the sense of Hyers–Ulam, Hyers–Ulam–Rassias, Mittag–Leffler–Hyers–Ulam and Mittag–Leffler–Hyers–Ulam–Rassias of n-th order linear differential equations are investigated. The main results are proved by applying generalized replacement lemma and generalized Gronwall’s inequality. The obtained results include many previous results and give improvements. In particular, the results have improved in two ways: no initial conditions for differential equations and no conditions for a coefficient. Most of the results are limited to the case where the interval is bounded, but a result for the unbounded case is obtained.

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References

  1. Anderson, D.R., Onitsuka, M.: Hyers–Ulam stability for differential systems with 2\(\times \)2 constant coefficient matrix. Results Math. 77, 136 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alqifiary, Q. H., Jung, S.-M.: Laplace transform and generalized Hyers–Ulam stability of linear differential equations. Electron. J. Differ. Equa., No. 80, 11pp (2014)

  3. Alsina, C., Ger, R.: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2(4), 373–380 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourgin, D.G.: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 57, 223–237 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brzdęk, J., Popa, D., Raşa, I., Xu, B.: Ulam Stability of Operators, A volume in Mathematical Analysis and its Applications, Academic Press, (2018)

  7. Cǎdariu, L., Popa, D., Raşa, I.: Ulam stability of a second linear differential operator with nonconstant coefficients. Symmetry 12(9), 1451 (2020)

    Article  Google Scholar 

  8. Collins, P.J.: Differential and Integral Equations. Oxford University Press, Oxford (2006)

    MATH  Google Scholar 

  9. Corduneanu, C.: Principles of Differential and Integral Equations. Chelsea Publ. Company, New York (1971)

    MATH  Google Scholar 

  10. Dragičević, D.: Hyers–Ulam stability for a class of perturbed Hill’s equations. Results Math. 76, 129 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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(2), 1432–1446 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gavruta, P., Jung, S.-M., Li, Y.: Hyers–Ulam stability for second-order linear differential equations with boundary conditions. Electron. J. Differ. Equ., No. 80, 5 pp (2011)

  13. Ghaemi, M.B., Bagher, M.E., Gordji, M.E., Alizadeh, B., Park, C.: Hyers–Ulam stability of exact second-order linear differential equations. Adv. Differ. Equ. 2012, 36 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hale, J.K.: Ordinary Differential Equations, 2nd edn. Robert E. Krieger Publishing Co., Inc, Huntington, N.Y. (1980)

    MATH  Google Scholar 

  15. Hyers, D.H.: On the stability of a linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17(10), 1135–1140 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jung, S.-M.: Hyers-Ulam stability of linear differential equations of first order, III. J. Math. Anal. Appl. 311(1), 139–146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. II. Appl. Math. Lett. 19(9), 854–858 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jung, S.-M.: Approximate solutions of a linear differential equation of third order. Bull. Malays. Math. Sci. Soc. 35(4), 1063–1073 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Jung, S.-M., Ponmana Selvan, A., Murali, R.: Mahgoub transform and Hyers–Ulam stability of first-order linear differential equations. J. Math. Inequal. 15(3), 1201–1218 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kalvandi, V., Eghbali, N., Rassias, J.M.: Mittag–Leffler–Hyers–Ulam stability of fractional differential equations of second order. J. Math. Ext. 13(1), 29–43 (2019)

    MATH  Google Scholar 

  22. Kanwal, R.P.: Linear Integral Equations. Theory & Technique. Reprint of the 2nd: edition, p. 2013. Modern Birkhäuser Classics, Birkhäuser/Springer, New York (1997)

  23. Li, Y., Shen, Y.: Hyers–Ulam stability of linear differential equations of second order. Appl. Math. Lett. 23(3), 306–309 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Miura, T.: On the Hyers–Ulam stability of a differentiable map. Sci. Math. Jpn. 55(1), 17–24 (2002)

    MathSciNet  MATH  Google Scholar 

  25. Miura, T., Jung, S.-M., Takahasi, S.-E.: Hyers–Ulam–Rassias stability of the Banach space valued linear differential equations \(y^{\prime } = \lambda y\). J. Korean Math. Soc. 41(6), 995–1005 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Miura, T., Miyajima, S., Takahasi, S.-E.: A characterization of Hyers–Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 286(1), 136–146 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Miura, T., Takahasi, S.-E., Choda, H.: On the Hyers–Ulam stability of real continuous function valued differentiable map. Tokyo J. Math. 24(2), 467–476 (2001)

    MathSciNet  MATH  Google Scholar 

  28. Modebei, M. I., Olaiya, O. O., Otaide, I.: Generalized Hyers-Ulam stability of second order linear ordinary differential equation with initial condition. Adv. Inequal. Appl. 2014, 36, 7 pp (2014)

  29. Murali, R., Ponmana Selvan, A.: Hyer–Ulam–Rassias stability for the linear ordinary differential equation of third order. Kragujevac J. Math. 42(4), 579–590 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Murali, R., Ponmana Selvan, A.: Hyers–Ulam stability of \(n\)th order linear differential equation. Proyecciones J. Math. (Revista de Matematica) 38(3), 553–566 (2019)

    MATH  Google Scholar 

  31. Murali, R., Ponmana Selvan, A., Park, C.: Ulam stability of linear differential equations using Fourier transform. AIMS Math. 5(2), 766–780 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Murali, R., Ponmana Selvan, A., Park, C., Lee, J.R.: Aboodh transform and the stability of second order linear differential equations. Adv. Differ. Equ. 2021, 296 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  33. Murali, R., Ponmana Selvan, A., Baskaran, S., Park C., Lee, J. R.: Hyers–Ulam stability of first-order linear differential equations using Aboodh transform. J. Inequal. Appl., Paper No. 133, 18 pp (2021)

  34. Obłoza, M.: Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. No. 13 , 259–270 (1993)

  35. Obłoza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. No. 14 , 141–146 (1997)

  36. Öğrekçi, S.: Stability of delay differential equations in the sense of Ulam on unbounded intervals. Int. J. Optim. Control. Theor. Appl. IJOCTA 9(2), 125–131 (2019)

    Article  MathSciNet  Google Scholar 

  37. Onitsuka, M., Shoji, T.: Hyers–Ulam stability of first-order homogeneous linear differential equations with a real-valued coefficient. Appl. Math. Lett. 63, 102–108 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Onitsuka, M.: Hyers–Ulam stability of first order linear differential equations of Carathéodory type and its application. Appl. Math. Lett. 90, 61–68 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. Popa, D., Raşa, I.: On the Hyers–Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381(2), 530–537 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Popa, D., Raşa, I.: Hyers–Ulam stability of the linear differential operator with nonconstant coefficients. Appl. Math. Comput. 219(4), 1562–1568 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ravi, K., Murali, R., Ponmana Selvan, A.: Hyers–Ulam stability of nth order linear differential equation with initial and boundary condition. Asian J. Math. Comput. Res. 11(3), 201–207 (2016)

    Google Scholar 

  43. Takahasi, S.-E., Miura, T., Miyajima, S.: On the Hyers–Ulam stability of the Banach space-valued differential equation \(y^{\prime }= \lambda y\). Bull. Korean Math. Soc. 39(2), 309–315 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Teschl, G.: Ordinary differential equations and dynamical systems. Graduate Studies in Mathematics, 140, American Mathematical Society, Providence, RI (2012)

  45. Ulam, S. M.: Problem in Modern Mathematics, Chapter IV, Science Editors, Willey, New York (1960)

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Funding

M.O. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant No. JP20K03668).

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Authors and Affiliations

  1. Department of Mathematics, Kings Engineering College, Sriperumbudur, Tamil Nadu, 602 117, India

    A. Ponmana Selvan

  2. Department of Applied Mathematics, Okayama University of Science, Okayama, 700-0005, Japan

    M. Onitsuka

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  1. A. Ponmana Selvan
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  2. M. Onitsuka
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Both authors contributed equally to the study conception and design. The first draft of the manuscript was written by APS and both authors commented on previous versions of the manuscript. Both authors read and approved the final manuscript.

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Correspondence to M. Onitsuka.

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Ponmana Selvan, A., Onitsuka, M. Ulam Type Stabilities of n-th Order Linear Differential Equations Using Gronwall’s Inequality. Results Math 78, 198 (2023). https://doi.org/10.1007/s00025-023-01975-7

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