The Banach fixed-point theorem states that if $f:[a,b]\to [a,b]$ is $\lambda$-Lipschitz where $\lambda\in[0,1)$ is such that satisfies $|f(x)-f(y)|\leq \lambda |x-y|$ for every $x,y\in [a,b]$ (I'm considering the restriction of the general version of the theorem to $(\mathbb R,|\cdot|)$), then there exists a unique fixed point $x^*\in [a,b]$, that is, $f(x^*)=x^*$ and can be found by taking an arbitrary $x_0\in[a,b]$ and defining $x_k = f(x_{k-1})$, so that $x_k\to x^*$ when $k\to \infty$.
However, in numerical analysis we are considering a more restrictive version (in order to determine when $f(x) = x$ and the solution $x$ is unique), which replaces the $\lambda$-Lipschitz condition by assuming that $|f'(x)| \leq \lambda < 1$ for some $\lambda$. This condition makes the theorem inapplicable when the functions vary with $|f'(x)| \geq 1$, so I would like to ask what conditions of the theorem do not rely on $|f'(x)| < 1$ but are stronger than the $\lambda$-Lipschitz. Is this associated to another theorem or idea?
Thanks in advance.
Consider that $f$ has a derivative (this is the case in the framework of most numerical analysis theorems) and such as $$ |f(x) - f(y)| \le \lambda |x-y| $$
Then when $y\to x$, using continuity of $x\to |x|$: $$ |f'(x)| = \left| \lim_{y\to x} \frac{f(x) - f(y)}{x-y} \right| =\lim_{y\to x} \frac{|f(x) - f(y)|}{|x-y|}\le \lambda $$
Note that on the other hand, the mean value theorem guarantees the other implication.