Norm additive mappings between the positive cones of continuous function algebras

We study bijections between the positive cones of spaces of continuous functions vanishing at infinity that satisfy a norm additive condition. Such maps arise naturally in the study of nonlinear functional equations and norm-preserving structures on function spaces. While in the compact (unital) case these maps can often be analyzed via linear extension techniques, the non-unital setting $C_0(X)$ requires a different approach due to the absence of a distinguished unit element. In this paper, we show that every bijection $T:C_0^{+}(X)\to C_0^{+}(Y)$ between the positive cones of $C_0(X)$ and $C_0(Y)$ satisfying $$\mathrm{\parallel }T(f+g)\mathrm{\parallel }=\mathrm{\parallel }Tf+Tg\mathrm{\parallel }$$ for all $f,g\in C_0^{+}(X)$ admits a representation of the form $$Tf(y)=h(y)f(\tau (y)),$$ where $\tau :Y\to X$ is a homeomorphism and $h$ is a bounded continuous function from $Y$ to $(0,\mathrm{\infty })$. This yields a complete characterization of norm additive bijections on positive cones of $C_0^{+}(X)$.