Stability of disjointness preserving mappings

Let $X$ and $Y$ be compact Hausdorff spaces and let $\varepsilon \ge 0$. A linear mapping $\Phi\colon\mathcal{C}(X) \to \mathcal{C}(Y)$ is called $\varepsilon$-disjointness preserving if $f g =0$ implies that $\Vert\Phi(f) \Phi(g)\Vert\le\varepsilon\Vert f\Vert \Vert g\Vert$. If $\Phi \colon \mathcal{C}(X) \to \mathcal{C}(Y)$ is a continuous or surjective $\varepsilon$-disjointness preserving linear mapping, we prove that there exists a disjointness preserving linear mapping $\Psi \colon \mathcal{C}(X) \to \mathcal{C}(Y)$ satisfying $\Vert\Phi(f)-\Psi(f)\Vert\le 20\sqrt{\varepsilon}\Vert f\Vert$. We also prove that every unbounded $\varepsilon$-disjointness preserving linear functional on $\mathcal{C}(X)$ is disjointness preserving.