More on Poincaré’s and Perron’s Theorems for Difference Equations∗

Abstract

Consider the scalar kth order linear difference equation: x(n + k) + pi(n)x(n + k - 1) + … + pk(n)x(n) = 0 where the limits qi=lim_n→∞Pi(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, ie., ϱ= lim sup_n→∞ |x(n)| is equal to the modulus of one of the roots of the characteristics equation χ ^k + q ₁χ ^k−1+…+qk=0. This result is a consequence of a more general theorem concerning the Poincaré difference system x(n+1)=[A+B(n]x(n), where A and B(n) (n=0,1,…) are square matrices such that ‖B(n)‖ →0 as n → ∞. As another corollary, we obtain a new limit relation for the solutions of (∗).