Instability of second-order nonhomogeneous linear difference equations with real-valued coefficients

Description

In J. Comput. Anal. Appl. (2020), pp. 152–165, the author dealt with Hyers–Ulam stability of the second-order linear difference equation $\Delta_h^2x(t)+\alpha \Delta_hx(t)+\beta x(t) = f(t)$ on $h\mathbb{Z}$ , where $\Delta_hx(t) = (x(t+h)-x(t))/h$ and $h\mathbb{Z} = \{hk|\,k\in\mathbb{Z}\}$ for the step size $h>0$ ; $\alpha$ and $\beta$ are real numbers; $f(t)$ is a real-valued function on $h\mathbb{Z}$ . The purpose of this paper is to clarify that the second-order linear difference equation has no Hyers–Ulam stability when the step size $h>0$ and the coefficients $\alpha$ and $\beta$ satisfy suitable conditions. Finally, a necessary and sufficient condition for Hyers–Ulam stability is obtained.