About combined non-expansive and potentially expansive properties of a class of self-maps in metric spaces

This paper investigates self-maps which satisfy a distance constraint in a metric space which mixed point-dependent non-expansive properties, or in particular contractive self-maps, and potentially expansive properties related to some distance threshold. In this way, the self-map may be expansive for small distances and contractive for large ones. The above mentioned constraint is feasible in certain real - world stability problems of usefulness as when discussing ultimate boundedness in dynamic systems.