Abstract
We show that, in the classes of functions with values in a real or complex Banach space, the problem of Hyers-Ulam stability of a linear functional equation of higher order (with constant coefficients) can be reduced to the problem of stability of a first order linear functional equation. As a consequence we prove that (under some weak additional assumptions) the linear equation of higher order, with constant coefficients, is stable in the case where its characteristic equation has no complex roots of module one. We also derive some results concerning solutions of the equation.
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Brzdęk, J., Popa, D. & Xu, B. Hyers-Ulam stability for linear equations of higher orders. Acta Math Hung 120, 1–8 (2008). https://doi.org/10.1007/s10474-007-7069-3
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DOI: https://doi.org/10.1007/s10474-007-7069-3