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+p−1 + · · · + a p−1 xn+1 + a p xn + bn , n ≥ 0, where x0 ,x1 ,.,x p−1 ∈ X. 1. Introduction In 1940, S. M. Ulam proposed the following problem. Problem . Given a metric group (G,. | 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 . ap e C and bn n 0 a sequence in X. We investigate the Hyers-Ulam stability of the linear recurrence xn p a Xn p-1 ap-1xn 1 apXn bn n 0 where x0 x1 . Xp-1 e X. 1. Introduction In 1940 S. M. Ulam proposed the following problem. Problem . Given a metric group G d a positive number e and a mapping f G G which satisfies the inequality d f xy f x f y e for all X y e G do there exist an automorphism a of G and a constant 8 depending only on G such that d a x f x 8 for all X e 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 spaces. This result represents the starting point theory of Hyers-Ulam stability of functional equations. Generally we say that a functional equation is stable in Hyers-Ulam sense if for every solution of the perturbed equation there exists a solution of the equation that differs from the solution of the perturbed equation with a small error. In the last 30 years the stability theory of functional equations was strongly developed. Recall that very important contributions to this subject were brought by Forti 2 Gavruta 3 Ger 4 Pales 6 7 Szekelyhidi 9 Rassias 8 and Trif 10 . As it is mentioned in 1 there are much less results on stability for functional equations in a single variable than in more variables and no surveys on this subject. In our paper we will investigate the discrete case for equations in single variable namely the Hyers-Ulam stability of linear recurrence with constant coefficients. Let X be a Banach space over a field K and xn p f Xn p-1 . xn n 0 a recurrence in X when p is a positive integer f Xp X is a mapping and x0 x1 . Xp-1 e X. We say that the recurrence is stable in
