Hyers-Ulam stability of K-Fibonacci functional equation

Let denote by F_{k,n} the n^{th} k-Fibonacci number where F_{k,n} = kF_{k,n-۱}+ F_{k,n-۲} for n\geq ۲ with initial conditions F_{k,۰} = ۰, F_{k,۱} = ۱, we may derive a functional equation f(k, x) = kf(k, x − ۱) + f(k, x − ۲). In this paper, we solve this equation and prove its Hyere-Ulam stability in the class of functions f : \mathbb{N}\times\mathbb{R}\to X, where X is a real Banach space.

Keywords:

stability , Fibonacci functional equation