The Discrete Diamond-Alpha Imaginary Ellipse and Hyers–Ulam Stability

Abstract

We introduce the imaginary diamond-alpha ellipse, which unifies and extends the left Hilger imaginary circle (forward, Delta case) and the right Hilger imaginary circle (backward, nabla case), for the discrete diamond-alpha derivative with constant step size. We then establish the Hyers–Ulam stability (HUS) of the firstorder linear complex constant coefficient discrete diamond-alpha derivative equation, proving that the imaginary diamond-alpha ellipse fails to have HUS, while inside and outside the ellipse the equation has HUS. In particular, for each parameter value not on the diamond-alpha ellipse, we determine explicitly the best (minimum) HUS constant in terms of the elliptical real part of the coefficient. AMS Subject Classifications: 30E10, 39A06, 39A30, 39A45.