Báo Cáo Báo cáo hóa học HYERS-ULAM STABILITY OF THE LINEAR RECURRENCE WITH CONSTANT COEFFICIENTS DORIAN POPA

HYERS-ULAM STABILITY OF THE LINEAR RECURRENCE WITH CONSTANT COEFFICIENTS DORIAN POPA Received 5 November 2004 and in revised form 14 March 2005

Let X be a Banach space over the field R or C, a1 , .,a p ∈ C, and (bn )n≥0 a sequence in X. We investigate the Hyers-Ulam stability of the linear recurrence xn+p = a1 xn+pư1 + · · · + a pư1 xn+1 + a p xn + bn , n ≥ 0, where x0 ,x1 , .,x pư1 ∈ X.

1. Introduction In 1940, S. M. Ulam proposed the following problem. Problem 1.1. Given a metric group (G, ·,d), a positive number ε, and a mapping f : G → G which satisfies the inequality d( f (xy), f (x) f (y)) ≤ ε for all x, y ∈ G, do there exist an automorphism a of G and a constant δ depending only on G such that d(a(x), f (x)) ≤ δ for all x ∈ G? If the answer to this question is affirmative, we say that the equation a(xy) = a(x)a(y) is stable. A first answer to this question was given by Hyers [5] in 1941 who proved that the Cauchy equation is stable in Banach