The dual Minkowski problem for positive indices

We derive the stability result of the dual curvature measure with near constant density in the even case. As an application, the existence and uniqueness of solutions to the even dual Minkowski problem for positive indices in ${\mathbb{R}}^{n+1}$ are obtained with $n\ge 1$, provided the density of the given measure is close to 1 in the $C^{\alpha }$ norm with $\alpha \in (0,1)$.