We prove the Hyers-Ulam stability of the Davison functional equation \[f(x+xy)+f(y)=f(x+y)+f(xy),\] for a class of mappings from a normed algebra $\mathcal{A}$ (with a unit element 1) into a Banach space $\mathcal{B}$, on the restricted domain $\left\{(x,y)\in \mathcal{A}×\mathcal{A}:~\min\{\|x\|,\|y\|\}\geqslant d\right\}$, where $d>0$ is a constant. As a result, we obtain some asymptotic behaviors of Davison mappings. In addition, we obtain the corollary that for every mapping $g$ from a normed algebra $\mathcal{A}$ into a normed space $\mathcal{B}$, and for all positive real numbers $r,s$, one of the following two conditions must be valid: \[up_{x,yı\mathcal{A}}eft\|g(x+y)+g(xy)-g(x+xy)-g(y)\right\|\cdot\|x\|^r\cdot\|y\|^s=+ıfty\] or \[g(x+y)+g(xy)=g(x+xy)+g(y).]