More on Poincare's and Perron's theorems for difference equations

Consider the scalar kth order linear difference equation x(n + k) + p(1)(n)x(n + k - 1) + ... + p(k)(n)x(n) = 0, (*) where the limits q(i) = lim(n-->infinity) p(i)(n) (i = 1, ...,k) are finite. In this paper, we confirm the conjecture formulated recently by Elaydi. Namely, every nonzero solution x of (*) satisfies the same asymptotic relation as the fundamental solutions described earlier by Perron, i.e., rho = lim sup(n-->infinity) (n)root\x(n)\ is equal to the modulus of one of the roots of the characteristic equation lambda(k) + q(1) lambda(k-1) + ... + q(k) = 0. This result is a consequence of a more general theorem concerning the Poincare difference system x(n + 1) = [A + B(n)]x(n), where A and B(n) (n = 0, 1,...) are square matrices such that parallel toB(n)parallel to --> 0 as n-->infinity. As another corollary, we obtain a new limit relation for the solutions of (*).