Suppose $F:(X,d) \to (X,d)$ a Lipschitz application with Lipschitz constant $L>1$. Is there a metric $\rho$ on $X$ sequentially equivalent to $d$ such that $F:(X,\rho)\to (X,\rho)$ is a $\lambda$-contraction?
The motivation for this question is the following. If $F: (X,\rho) \to (X,\rho)$ is a contraction then by Banach's fixed point theorem $F: (X,\rho) \to (X,\rho)$ has a fixed point at $(X,\rho)$. And since $\rho$ is sequentially equivalent to $d$ then $F:(X,d)\to (X,d)$ has a fixed point at (X,d). If this motivation is correct and the answer to the question is affirmative, then the conclusion is that every Lipschitz application of a metric space itself has a single fixed point.
But I believe that such a conclusion is not possible because if it were, analysis textbooks would not restrict the Banach's Fixed Point Theorem to contractive applications.
QUESTION. Any explanation for any negative answer to the above question? If the answer to the question is yes, any explanation for why the Analysis textbooks do not explore the fixed-point result mentioned above?