I have a fixed-point question similar to the Banach fixed-point theorem.
Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a given function and let $x^\star \in \mathbb{R}^n$ be a known fixed-point of $f$, i.e. $f(x^\star) = x^\star$.
It is also known that $f$ satisfies: \begin{eqnarray} \sup_{y\neq0} \frac{\|f(x^\star+y) - f(x^\star) \|_2}{\|y\|_2} \leq \alpha. \hspace{1in} (1) \end{eqnarray} for some value of $\alpha \in (0,1)$. Can we conclude that (1) $x^\star$ is the only fixed-point of $f$ and (2) that iterating $f$ will yield $x^\star$.
If (1) was true for all $x$ rather just $x^\star$ then $f$ would be Lipschitz and a contraction so $x^\star$ would be unique (and not needed to be assumed) and iterating $f$ would yield it. However, (1) assumes less about the continuity of $f$ but does bound it to a cone with slope $\alpha<1$ around $x^\star$. Is this enough or where does it fall apart?