Hyers-Ulam stability of the linear recurrence with constant coefficients
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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 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 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], G˘avrut¸a [3], Ger [4], P´ales [6, 7], Sz´ekelyhidi [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,
(1.1)
a recurrence in X, when p is a positive integer, f : X p → X is a mapping, and x0 ,x1 ,...,x p−1 ∈ X. We say that the recurrence (1.1) is stable in Hyers-Ulam sense if for every positive ε Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:2 (2005) 101–107 DOI: 10.1155/ADE.2005.101
102
Hyers-Ulam stability of a linear recurrence
and every sequence (xn )n≥0 that satisfies the inequality xn+p − f xn+p−1 ,...,xn < ε,
n ≥ 0,
(1.2)
there exist a sequence (yn )n≥0 given by the recurrence (1.1) and a positive δ depending only on f such that xn − yn < δ,
n ≥ 0.
(1.3)
In [7], the author investigates the Hyers-Ulam-Rassias stability of the first-order linear recurrence in a Banach space. Using some ideas from [7] in this paper, one obtains a result concerning the stability of the n-order linear recurrence with constant coefficients in a Banach space, namely, xn+p = a1 xn+p−1 + · · · + a p−1 xn+1 a + a p xn + bn ,
n ≥ 0,
(1.4)
where a1 ,a2 ,...,a p ∈ K, (bn )n≥0 is a given sequence in X, and x0 ,x1 ,...,x p−1 ∈ X. Many new and interesting results concerning difference equations can be found in [1]. 2. Main results In what follows, we denote by K the
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