An open mapping theorem for nonlinear operator equations associated with elliptic complexes

Let $\{A^i,E^i\}$ be the elliptic complex on a $n$-dimensional smooth closed Riemannian manifold $X$ with the first order differential operators $A^i$ and smooth vector bundles $E^i$ over $X$. We consider nonlinear operator equations, associated with the parabolic differential operators $\partial_t + \Delta^{i}$, generated by the Laplacians $\Delta^{i}$ of the complex $\{A^i,E^i\}$, in special Bochner-Sobolev functional spaces. We prove that under reasonable assumptions regarding the nonlinear term the Frech\'et derivative $\mathcal{A}_i'$ of the induced nonlinear mapping is continuously invertible and the map $\mathcal{A}_i$ is open and injective in chosen spaces.