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

 Onitsuka, Masakazu

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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.

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 Onitsuka, Masakazu