Let $E$ a Banach's space and $X\subset E$ open. The Banach's homeomorphism theorem tells us that if a function $F:X\to E$ is a contraction on $X$ then $(I+F):X\to E$ is a homeomorphism of $X$ onto the open subset $(I+F)(X)$ of $E$, where $I$ denotes the identity map on $E$.
The proof most commonly of the Banach's homeomorphism theorem seen in textbooks of relies on the Banach fixed-point theorem.
I ask, is there any proof of Banach's theorem homeomorphism that do not use the contraction map principle? Some reference?
I have thought of a proof in spaces of finite demension. But in the midst of my attempts to end the proof, I end up arriving in places where the only output I see to complete the proof is to apply some equivalent idea to contractiom map principle or the domain invariance theorem. And I want to avoid these two results.