Generalized Hyers-Ulam stability of the second-order linear differential equations. (English) Zbl 1241.34014

Summary: We prove generalized Hyers-Ulam stability for the second-order linear differential equation \[ y'' + p(x)y' + q(x)y = f(x) \] under the condition that there exists a nonzero \(y_1 : I \rightarrow X\) in \(C^2(I)\) such that \(y_1'' + p(x)y_1' + q(x)y_1 = 0\) and \(I\) is an open interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam stability for several important well-known differential equations.


34A30 Linear ordinary differential equations and systems
34D99 Stability theory for ordinary differential equations
Full Text: DOI


[1] S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1940. · Zbl 0137.24201
[2] D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 · doi:10.2307/2042795
[4] C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential function,” Journal of Inequalities and Applications, vol. 2, no. 4, pp. 373-380, 1998. · Zbl 0918.39009 · doi:10.1155/S102558349800023X
[5] S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation y\(^{\prime}\)=\lambda y,” Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309-315, 2002. · Zbl 1011.34046 · doi:10.4134/BKMS.2002.39.2.309
[6] T. Miura, S. Miyajima, and S.-E. Takahasi, “A characterization of Hyers-Ulam stability of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 136-146, 2003. · Zbl 1045.47037 · doi:10.1016/S0022-247X(03)00458-X
[7] S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1135-1140, 2004. · Zbl 1061.34039 · doi:10.1016/j.aml.2003.11.004
[8] S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. II,” Applied Mathematics Letters, vol. 19, no. 9, pp. 854-858, 2006. · Zbl 1125.34328 · doi:10.1016/j.aml.2005.11.004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.