Hyers-ulam Stability of the Linear Recurrence with Constant Coefficients

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], Găvruţa [3], Ger [4], Páles [6, 7], Székelyhidi [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