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Laplace transform and Hyers-Ulam stability of linear differential equations. (English) Zbl 1286.34077
Summary: We prove the Hyers-Ulam stability of a linear differential equation of the \(n\)th order. More precisely, applying the Laplace transform method, we prove that the differential equation \[ y^{(n)}(t)+\sum_{k=0}^{n-1}\alpha_ky^{(k)}(t)=f(t) \] has Hyers-Ulam stability, where \(\alpha_k\) is a scalar, \(y\) and \(f\) are \(n\) times continuously differentiable and of exponential order, respectively.

MSC:
34D10 Perturbations of ordinary differential equations
34A30 Linear ordinary differential equations and systems, general
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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[1] Alsina, C.; Ger, R., On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 2, 373-380, (1998) · Zbl 0918.39009
[2] Cimpean, D. S.; Popa, D., On the stability of the linear differential equation of higher order with constant coefficients, Appl. Math. Comput., 217, 4141-4146, (2010) · Zbl 1211.34065
[3] Czerwik, S., Functional equations and inequalities in several variables, (2002), World Scientific Singapore · Zbl 1011.39019
[4] Davies, B., Integral transforms and their apllications, (2001), Springer New York
[5] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27, 222-224, (1941) · JFM 67.0424.01
[6] Hyers, D. H.; Isac, G.; Rassias, Th. M., Stability of functional equations in several variables, (1998), Birkhäuser Boston · Zbl 0894.39012
[7] Jung, S.-M., Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17, 1135-1140, (2004) · Zbl 1061.34039
[8] Jung, S.-M., Hyers-Ulam stability of linear differential equations of first order, III, J. Math. Anal. Appl., 311, 139-146, (2005) · Zbl 1087.34534
[9] Jung, S.-M, Hyers-Ulam stability of linear differential equations of first order, II, Appl. Math. Lett., 19, 854-858, (2006) · Zbl 1125.34328
[10] Jung, S.-M., Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, (2011), Springer New York · Zbl 1221.39038
[11] Li, Y.; Shen, Y., Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett., 23, 306-309, (2010) · Zbl 1188.34069
[12] Miura, T.; Miyajima, S.; Takahasi, S. E., A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl., 286, 136-146, (2003) · Zbl 1045.47037
[13] Miura, T.; Miyajima, S.; Takahasi, S. E., Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr., 258, 90-96, (2003) · Zbl 1039.34054
[14] Obłoza, M., Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat., 13, 259-270, (1993) · Zbl 0964.34514
[15] Obłoza, M., Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat., 14, 141-146, (1997) · Zbl 1159.34332
[16] Popa, D.; Raşa, I., On the Hyers-Ulam stability of the linear differential equation, J. Math. Anal. Appl., 381, 530-537, (2011) · Zbl 1222.34069
[17] Popa, D.; Raşa, I., Hyers-Ulam stability of the linear differential operator with non-constant coefficients, Appl. Math. Comput., 219, 1562-1568, (2012) · Zbl 1368.34075
[18] Rus, I. A., Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10, 305-320, (2009) · Zbl 1204.47071
[19] Rus, I. A., Ulam stability of ordinary differential equations, Stud. Univ. Babes-Bolyai Math., 54, 125-134, (2009) · Zbl 1224.34165
[20] 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, 309-315, (2002) · Zbl 1011.34046
[21] Takahasi, S. E.; Takagi, H.; Miura, T.; Miyajima, S., The Hyers-Ulam stability constants of first order linear differential operators, J. Math. Anal. Appl., 296, 403-409, (2004) · Zbl 1074.47022
[22] Ulam, S. M., (A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, No. 8, (1960), Interscience New York) · Zbl 0086.24101
[23] Wang, G.; Zhou, M.; Sun, L., Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 21, 1024-1028, (2008) · Zbl 1159.34041
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