### Book chapter

## Modular Lipschitzian and contractive maps

In the context of metric modular spaces, introduced recently by the author, we define the notion of modular Lipschitzian maps between modular spaces, as an extension of the notion of Lipschitzian maps between metric spaces, and address a modular version of Banach’s Fixed Point Theorem for modular contractive maps. We show that the assumptions in our fixed point theorem are sharp and that it guarantees the existence of fixed points in cases when Banach’s Theorem is inapplicable.