J. Math. Anal. Appl. 309 (2005) 591–597 www.elsevier.com/locate/jmaa Hyers–Ulam–Rassias stability of a linear recurrence Dorian Popa Technical University, Department of Mathematics, Str. C. Daicoviciu, 15, Cluj-Napoca, Romania Received 27 April 2004 Available online 1 April 2005 Submitted by R.P. Agarwal Abstract In this paper we give a Hyers–Ulam–Rassias stability result for the first order linear recurrence in Banach spaces. 2004 Elsevier Inc. All rights reserved. Keywords: Hyers–Ulam–Rassias stability; Recurrence; Sequence 1. Introduction In 1940 S.M. Ulam proposed the following problem: 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, does 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 D.H. Hyers [3,4] in 1941 who proved that the Cauchy equation is stable in Banach spaces. This is the starting point of the Hyers– Ulam stability theory of functional equations. Th.M. Rassias [7–9] introduced a new notion E-mail address: Popa.Dorian@math.utcluj.ro. 0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.10.013