Best Constant for Hyers–Ulam Stability of Second-Order h-Difference Equations with Constant Coefficients
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First, we prove that the best (minimum) constant for Hyers–Ulam stability of first-order linear h-difference equations with a complex constant coefficient is the reciprocal of the absolute value of the Hilger real part of that coefficient; if the coefficient lies on the Hilger imaginary circle, then the equation is unstable in the Hyers–Ulam sense. Second, using this best constant from the first-order complex coefficient case, we determine the best constant for Hyers–Ulam stability of second-order linear h-difference equations with constant real coefficients. The second-order equation also is Hyers–Ulam stable if and only if the values of the characteristic equation do not intersect the Hilger imaginary circle.
KeywordsStability first-order second-order Hyers–Ulam difference equations variation of constants constant coefficients Hilger imaginary circle Hilger real part
Mathematics Subject Classification39A10 34A30 34N05 39A30
The second author was supported by JSPS KAKENHI Grant Number JP17K14226.
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