Stability of an Abel functional equation over restricted domains

Let \(\mathbb R\) $\mathbb{R}$ be the set of real numbers and Y a Banach space. In this work, we study the Hyers-Ulam stability for the Abel functional equation \(\begin{aligned} f(x+y)=g(xy)+h(x-y),\;\; x,y\in \mathbb R \end{aligned}\) $$\begin{array}{c}\hfill f(x+y)=g(xy)+h(x-y),x,y\in \mathbb{R}\end{array}$$ where \(f,g,h:\mathbb R\rightarrow Y\) $f,g,h:\mathbb{R}\to Y$ are unknown functions. The perturbation of above equation in the restricted domains are also proved. As a direct consequence we obtain asymptotic behaviors of the proposed equation.