AIMS Mathematics, 2020, 5(2): 766-780. doi: 10.3934/math.2020052

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Ulam stability of linear differential equations using Fourier transform

1 PG and Research Department of Mathematics, Sacred Heart College, Tirupattur-635601, Vellore Dist., Tamil Nadu, India
2 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

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The purpose of this paper is to study the Hyers-Ulam stability and generalized HyersUlam stability of general linear differential equations of nth order with constant coefficients by using the Fourier transform method. Moreover, the Hyers-Ulam stability constants are obtained for these differential equations.
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# References

1. Q. H. Alqifiary, S. Jung, Laplace transform and generalized Hyers-Ulam stability of linear differential equations, Electron. J. Differ. Eq., 2014 (2014), 1-11.

2. Q. H. Alqifiary, J. K. Miljanovic, Note on the stability of system of differential equations $\dot{x}(t)= f(t, x(t))$, Gen. Math. Notes, 20 (2014), 27-33.

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

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

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

6. D. H. Hyers, On the stability of a linear functional equation, P. Natl. Acad. Sci. USA., 27 (1941), 222-224.

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

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

9. S. M. Jung, On the quadratic functional equation modulo a subgroup, Indian J. Pure Appl. Math., 36 (2005), 441-450.

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

11. S. M. Jung, Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl., 320 (2006), 549-561.

12. S. M. Jung, Approximate solution of a linear differential equation of third order, B. Malays. Math. Sci. So., 35 (2012), 1063-1073.

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

14. T. Li, A. Zada, S. Faisal, Hyers-Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl., 9 (2016), 2070-2075.

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

16. K. Liu, M. Feckan, D. O'Regan, et al. Hyers-Ulam stability and existence of solutions for differential equations with Caputo-Fabrizio fractional derivative, Mathematics, 7 (2019), 333.

17. T. Miura, S. Jung, S. E. Takahasi, Hyers-Ulam-Rassias stability of the Banach space valued linear differential equation $y^{'} = \lambda y$, J. Korean Math. Soc., 41 (2004), 995-1005.

18. R. Murali, A. P. Selvan, On the generalized Hyers-Ulam stability of linear ordinary differential equations of higher order, Int. J. Pure Appl. Math., 117 (2017), 317-326.

19. M. Ramdoss, P. S. Arumugan, Fourier transforms and Ulam stabilities of linear differential equations, In: G. Anastassiou, J. Rassias, editors, Frontiers in Functional Equations and Analytic Inequalities, Springer, Cham, 2019, 195-217.

20. M. Obloza, Hyers stability of the linear differential equation, Rockznik Nauk-Dydakt. Prace Mat., 13 (1993), 259-270.

21. M. Obloza, Connection between Hyers and Lyapunov stability of the ordinary differential equations, Rockznik Nauk-Dydakt. Prace Mat., 14 (1997), 141-146.

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

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

24. T. M. Rassias, On the stability of the linear mappings in Banach spaces, P. Am. Math. Soc., 72 (1978), 297-300.

25. I. A. Rus, Ulam stabilities of ordinary differential equations in Banach space, Carpathian J. Math., 26 (2010), 103-107.

26. S. E. Takahasi, T. Miura, S. Miyajima, On the Hyers-Ulam stability of the Banach space-valued differential equation $y'= \alpha y$, Bull. Korean Math. Soc., 39 (2002), 309-315.

27. S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960.

28. G. Wang, M. Zhou, L. Sun, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 21 (2008), 1024-1028.

29. J. R. Wang, A. Zada, W. Ali, Ulam's type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces, Int. J. Nonlin. Sci. Num., 19 (2018), 553-560.

30. X. Wang, M. Arif, A. Zada, β-Hyers-Ulam-Rassias stability of semilinear nonautonomous impulsive system, Symmetry, 11 (2019), 231.

31. A. Zada, W. Ali, C. Park, Ulam's type stability of higher order nonlinear delay differential equations via integral inequality of Gronwall Bellman-Bihari's type, Appl. Math. Comput., 350 (2019), 60-65.

32. A. Zada, S. O. Shah, Hyers-Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses, Hacet. J. Math. Stat., 47 (2018), 1196-1205.

33. A. Zada, S. Shaleena, T. Li, Stability analysis of higher order nonlinear differential equations in β-normed spaces, Math. Method. Appl. Sci., 42 (2019), 1151-1166.

34. A. Zada, M. Yar, T. Li, Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions, Ann. Univ. Paedagog. Crac. Stud. Math., 17 (2018), 103-125.