Let $T: \mathtt{D} \to \mathtt{D}$ be a Lipschitz continuous operator, where $\mathtt{D} \subset \mathbb{R}^n$ is convex and compact. In particular, $\|T(x) - T(y)\| \leqslant L \| x - y \|$ for all $x, y \in \mathbb{R}^n$ and positive constant $L>0$. Since $\mathtt{D}$ is convex and compact, Brouwer fixed point theorem guarantees existence of fixed point of $T$ i.e. there exists $x_0 \in \mathtt{D}$ such that $T(x_0) = x_0$.
When $L<1$, $T$ is contraction and banach fixed point theorem proves that Picard iterations converge. When $L \leqslant 1$, T is nonexpansive and Mann iterations provide convergence results (under some additional assumptions). But what about the case $L>1$ (Lipschitz constant is strictly greater than 1)? Does there exists any iterative algorithm to approximate the fixed point of $T$? To be precise, does there exists an algorithm which produce sequence of iterates which converge to any fixed point of $T$?