I'm trying to prove the following assertion.
Hypothesis:
$1)$ $X$ is a Banach space and $Y$ is a normed space.
$2)$ $A : X → Y$ is a bounded bijective operator and $A^{-1}$ is bounded.
$3)$ $B : X → Y$ is a bounded operator and $∥A^{-1}(B-A)∥<1$.
Thesis:
$B$ is bijective and $\Vert B^{-1}∥ ≤ \frac{∥A^{-1}∥}{1-∥A^{-1}(B-A)∥}$
Now, if $\Vert L \Vert < 1$ then $(I-L)$ is bijective, so $ I – A^{-1}(B-A) = -A^{-1}B$ is bijective. But $(-A^{-1})$ is obviously bijective, so $B$ is also bijective. I need a little help for the bound.
Let $T=A^{-1}(B-A)$. Then $(I-T)^{-1}=\sum\limits_{n=0}^{\infty} T^{n}$, so $\|(I-T)^{-1}\| \leq \sum\limits_{n=0}^{\infty} \|T^{n}\| =\frac 1 {1-\|T\|}$ and the result follows from the fact that $B^{-1}=-(I-T)^{-1}A^{-1}$