Auslander's formula for contravariantly finite subcategories

Let $A$ be a right coherent ring and $\mathcal{X}$ contravariantly finite subcategory of ${\rm{mod}}\mbox{-}A$ containing projectives. In this paper, we construct recollement abelian categories $({\rm{mod}}_{0}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}A)$, where ${\rm{mod}}_{0}\mbox{-}\mathcal{X}$ is full ${\rm{mod}}\mbox{-}\mathcal{X}$ consisting all functors vanishing on projective modules. As result, relative version Auslander's formula with respect to will given. Some examples applications provided.