What is the relationship between contractive and Lipschitz functions?

[KF33]

I was looking at this definition of a contractive function and the only difference I saw between it and a Lipschitz function was the b and M. I am just wondering how you look at the connections between them.

 

[wrobel]

for contraction mapping the main thing is that $b<1$. Moreover, contraction takes a metric space into itself. Each contraction is a Lip. mapping