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We investigate the approximate solutions y:[−τ,∞)→Rof the delay differential equation y'(t)=λy(t-τ)(t∈[0,∞))with an initial condition, where λ>0and τ>0are 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

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Hyers-Ulam Stability of the Delay Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>y</mml:mi><mml:mi mathvariant="bold">'</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>y</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:math>

Hyers-Ulam Stability of the Delay Equationy'(t)=λy(t-τ)