# Hyers--Ulam stability of nth order linear differential equations

Volume 9, Issue 5, pp 2070--2075 Publication Date: May 20, 2016
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### Authors

Tongxing Li - School of Informatics, Linyi University, Linyi, Shandong 276005, P. R. China. - LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P. R. China. Akbar Zada - Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan. Shah Faisal - Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.

### Abstract

For nth order linear homogeneous and nonhomogeneous differential equations with nonconstant coefficients, we prove Hyers{Ulam stability by using open mapping theorem. The generalized Hyers{Ulam stability is also investigated.

### Keywords

• Hyers-Ulam stability
• generalized Hyers-Ulam stability
• nth order linear differential equation
• open mapping theorem.

•  35B35

### References

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

• [2] T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.

• [3] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951), 223-237.

• [4] J. Brzdęk, D. Popa, B. Xu , Remarks on stability of linear recurrence of higher order, Appl. Math. Lett., 23 (2010), 1459-1463.

• [5] M. Burger, N. Ozawa, A. Thom , On Ulam stability, Israel J. Math., 193 (2013), 109-129.

• [6] G. Choi, S.-M. Jung, Invariance of Hyers-Ulam stability of linear differential equations and its applications, Adv. Difference Equ., 2015 (2015), 14 pages.

• [7] J. Chung, Hyers-Ulam stability theorems for Pexider equations in the space of Schwartz distributions , Arch. Math., 84 (2005), 527-537.

• [8] J. Huang, S.-M. Jung, Y. Li , On Hyers-Ulam stability of nonlinear differential equations , Bull. Korean Math. Soc., 52 (2015), 685-697.

• [9] J. Huang, Y. Li, Hyers-Ulam stability of linear functional differential equations, J. Math. Anal. Appl., 426 (2015), 1192-1200.

• [10] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224.

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

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

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

• [14] Y. Li, Y. Shen, Hyers-Ulam stability of nonhomogeneous linear differential equations of second order, Int. J. Math. Math. Sci., 2009 (2009), 7 pages.

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

• [16] T. Miura, S. Miyajima, S.-E. Takahasi, A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl., 286 (2003), 136-146.

• [17] M. Ob loza , Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat., 13 (1993), 259-270.

• [18] M. Obloza , Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat., 14 (1997), 141-146.

• [19] D. Popa, Hyers-Ulam-Rassias stability of a linear recurrence, J. Math. Anal. Appl., 309 (2005), 591-597.

• [20] D. Popa, I.Raşa , Hyers-Ulam stability of the linear differential operator with nonconstant coeficients, Appl. Math. Comput., 219 (2012), 1562-1568.

• [21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces , Proc. Amer. Math. Soc., 72 (1978), 297-300.

• [22] H. Rezaei, S.-M. Jung, Th. M. Rassias , Laplace transform and Hyers-Ulam stability of linear differential equations, J. Math. Anal. Appl., 403 (2013), 244-251.

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

• [24] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York (1960)

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

• [26] A. Zada, O. Shah, R. Shah, Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput., 271 (2015), 512-518.

• [27] Z. Zheng , Theory of Functional Differential Equations, Anhui Education Press, in Chinese (1994)

• [28] D. G. Zill, A First Course in Differential Equations with Modeling Applications, Brooks/Cole, USA (2012)