A variant of Tingley's problem for positive unit spheres of continuous functions that vanish at infinity

Let $S(C_0(X){)}^{+}$ and $S(C_0(Y){)}^{+}$ denote the positive parts of the unit spheres of $C_0(X)$ and $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces. We prove that every surjective isometry from $S(C_0(X){)}^{+}$ onto $S(C_0(Y){)}^{+}$ is a composition operator induced by a homeomorphism between $X$ and $Y$ . As a consequence, such a map extends to a surjective reallinear isometry from $C_0(X)$ onto $C_0(Y)$. We also characterize surjective phase-isometries on the positive unit sphere.