Abstract and Applied Analysis
Volume 2010 (2010), Article ID 372176, 10 pages
doi:10.1155/2010/372176
Research Article

Hyers-Ulam Stability of the Delay Equation 𝑦 ′ ( 𝑡 ) = 𝜆 𝑦 ( 𝑡 − 𝜏 )

Soon-Mo Jung1 and Janusz Brzdęk2

1Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, Republic of Korea
2Department of Mathematics, Pedagogical University of Cracow, Podchorążych 2, 30-084 Kraków, Poland

Received 11 September 2010; Accepted 25 November 2010

Academic Editor: Yong Zhou

Copyright © 2010 Soon-Mo Jung and Janusz Brzdęk. 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 investigate the approximate solutions 𝑦 ∶ [ − 𝜏 , ∞ ) → ℝ of the delay differential equation 𝑦 ( 𝑡 ) = 𝜆 𝑦 ( 𝑡 − 𝜏 ) ( 𝑡 ∈ [ 0 , ∞ ) ) with an initial condition, where 𝜆 > 0 and 𝜏 > 0 are real constants. We show that they can be “approximated” by solutions of the equation that are constant on the interval [ − 𝜏 , 0 ] and, therefore, have quite simple forms. Our results correspond to the notion of stability introduced by Ulam and Hyers.