Hyers-Ulam stability of k-Fibonacci functional equation

Let denote by Fk,n the nth k-Fibonacci number where Fk,n = kFk,n?1+ Fk,n?2 for n  2 with initial conditions Fk,0 = 0, Fk,1 = 1, we may derive a functional equation f(k, x) = kf(k, x ? 1) + f(k, x ? 2). In this paper, we solve this equation and prove its Hyere-Ulam stability in the class of functions f : N×R ! X, where X is a real Banach space.