Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients

Abstract EN

We study the Hyers-Ulam stability in a Banach space X of the system of first order linear difference equations of the form xn+1=Axn+dn for n∈N0 (nonnegative integers), where A is a given r×r matrix with real or complex coefficients, respectively, and (dn)n∈N0 is a fixed sequence in Xr.

That is, we investigate the sequences (yn)n∈N0 in Xr such that δ∶=supn∈N0yn+1-Ayn-dn<∞ (with the maximum norm in Xr) and show that, in the case where all the eigenvalues of A are not of modulus 1, there is a positive real constant c (dependent only on A) such that, for each such a sequence (yn)n∈N0, there is a solution (xn)n∈N0 of the system with supn∈N0yn-xn≤cδ.