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

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