According to the Banach theorem on the inverse operator, for the continuous invertibility of a linear continuous operator acting from one Banach space to another, it is necessary and sufficient that this operator be surjective and injective. These requirements, even in the case of nonlinear operators, are sufficient for the invertibility of the corresponding operators. However, inverse operators may not be continuous. This applies to differentiable mappings for which inverse mappings may not be differentiable. Therefore, for the reversibility of nonlinear operators it is necessary to fulfill additional requirements. For a differentiable mapping, such a requirement is that the condition for the non-degeneracy of the Frechet derivative of the mapping at every point in the space in which this mapping acts is satisfied.
The article considers nonlinear autonomous differential operators of class $C^1$ that act from the space of bounded and continuously differentiable on the axis of functions to the space of bounded and continuous on the axis of functions with values in an infinite-dimensional Banach space. For such operators, necessary and sufficient conditions are given under which these operators are diffeomorphisms of the class $C^1$. The conditions of injectivity and surjectivity of the studied operators are also given.

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