I'm not so sure, but you can find an answer to your question in ROBERT H. MARTIN, NONLINEAR OPERATORS AND DIFFERENTIAL EQUATIONS IN BANACH SPACES by JOHN WILEY & SONS, 1976

Dear Yaroslav, in some particular cases, it exists theorems of type 1 or 2 for nonlinear operators. But only for "very particular" cases!

The results of the type

E, F Banach spaces; T an operator(not necessarily linear) from E to F satisfying some properties. Then T is invertible( or T is invertible and the inverse of T is continuous; or T is surjective)

are more common, because they have applications in the study of some nonlinear equations(existence, uniqueness and properties of solution). I published many papers about, consisting in obtaining some theorems of this type, followed by applications to concrete nonlinear equations. I attach some papers. For more details you can look at the papers from my researchgate account.

Good luck in your research!

Dear Professor,

You may want to consult the following manuscript:

Preprint On The Development of Nonlinear Operator Theory

Regards