Journal of Applied Mathematics
Volume 2011 (2011), Article ID 813137, 10 pages
http://dx.doi.org/10.1155/2011/813137
Research Article

Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations

A. Javadian,1 E. Sorouri,2 G. H. Kim,3 and M. Eshaghi Gordji2

1Department of Physics, Semnan University, P. O. Box 35195-363, Semnan, Iran
2Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran
3Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 26 September 2011; Accepted 23 October 2011

Academic Editor: Kuppalapalle Vajravelu

Copyright © 2011 A. Javadian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential equation of the form 𝑦 + 𝑝 ( 𝑥 ) 𝑦 + 𝑞 ( 𝑥 ) 𝑦 = 𝑓 ( 𝑥 ) , with condition that there exists a nonzero 𝑦 1 ∶ 𝐼 → 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 of several important well-known differential equations.