Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues.

Let q >= 2 be a positive integer and let (aj),(bj) and (cj) (with j nonnegative integer) be three given C-valued and q-periodic sequences. Let A(q):=Aq-1MIDLINE HORIZONTAL ELLIPSISA0, where Aj is defined below. Assume that the eigenvalues x,y,z of the "monodromy matrix" A(q) verify the condition (x-y)(y-z)(z-x)not equal 0. We prove that the linear recurrence in C xn+3=anxn+2+bnxn+1+cnxn,n is an element of Z+ is Hyers-Ulam stable if and only if (|x|-1)(|y|-1)(|z|-1)not equal 0, i.e., the spectrum of A(q) does not intersect the unit circle Gamma:={w is an element of C:|w|=1}.