Lipschitz inverse mapping theorem

Let $(E,\parallel \cdot \parallel )$ be a Banach space and let $A:E\to E$ be abounded linear isomorphism withbounded inverse (i.e. a topological linear automorphism);let $B(r)$ be the ball with center 0and radius $r$ (we allow $r=\mathrm{\infty }$). Then for any Lipschitz map$\varphi :B(r)\to E$such that and $\varphi (0)=0$, there are open sets$U\subset E$ and $V\subset B(r)$ and a map $T:U\to V$ such that $T(A+\varphi )={I|}_V$ and $(A+\varphi )T={I|}_U$.In other words, there is a local inverse of $A+\varphi$ near zero. Furthermore, the inverse $T$ is Lipschitz with $\mathrm{Lip}T\le {(\parallel A\parallel +\mathrm{Lip}\varphi )}^{-1}$ and

$$B\left(r({\parallel A^{-1}\parallel }^{-1}-\mathrm{Lip}\varphi )\right)\subset U.$$

Remark. The inclusion above implies that $A+\varphi :E\to E$ is invertible if $r=\mathrm{\infty }$.

Remark. $\mathrm{Lip}\varphi$ denotes the smallest Lipschitz constant of $\varphi$.