Note on nonstability of the linear functional equation of higher order

Authors:
Janusz Brzdek;Dorian Popa;Bing Xu
Affiliations:
Department of Mathematics, Pedagogical University, Podchoraych2, PL-30-084 Kraków, Poland;Department of Mathematics, Technical University, Str. C. Daicoviciu 15, Cluj-Napoca, 400020, Romania;Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
Venue:
Computers & Mathematics with Applications
Year:
2011

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Abstract

We provide a complete solution of the problem of Hyers-Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1. Our results are related to the notions of shadowing (in dynamical systems and computer science) and controlled chaos. They also correspond to some earlier results on approximate solutions of functional equations in single variable.