Implicit Function Theorem for Lipschitz Contractions

Theorem
Let $M$ and $N$ be metric spaces.

Let $M$ be complete.

Let $f : M \times N \to M$ be a Lipschitz continuous uniform contraction.

Then for all $t\in N$ there exists a unique $\map g t \in M$ such that $\map f {\map g t, t} = \map g t$, and the mapping $g : N \to M$ is Lipschitz continuous.

Proof
For every $t\in N$, the mapping:
 * $f_t : M \to M : x \mapsto \map f {x, t}$ is a contraction mapping.

By the Banach Fixed-Point Theorem, there exists a unique $\map g t \in M$ such that $\map {f_t} {\map g t} = \map g t$.

We show that $g$ is Lipschitz continuous.

Let $K<1$ be a uniform Lipschitz constant for $f$.

Let $L$ be a Lipschitz constant for $f$.

Let $s,t\in N$.

Then

and thus:

Thus $g$ is Lipschitz continuous.