Hyers-Ulam Stability of the Delay Equation y ' ( t ) = λ y ( t - τ )

Abstract

We investigate the approximate solutions $y:[-\tau ,\infty )\to ℝ$ of the delay differential equation $y\text{'}\left(t\right)=\lambda y(t-\tau )(t\in [0,\infty ))$ with an initial condition, where $\lambda >0$ and $\tau >0$ are real constants. We show that they can be “approximated” by solutions of the equation that are constant on the interval $[-\tau ,0]$ and, therefore, have quite simple forms. Our results correspond to the notion of stability introduced by Ulam and Hyers.