  We introduce a version of the Omori-Yau maximum principle which generalizes
the version obtained by Pigola-Rigoli-Setti 21. We apply our method to derive a
non-trivial generalization Jorge-Koutrofiotis Theorem 15 for cylindrically
bounded submanifolds due to Alias-Bessa-Montenegro 2, we extend results due to
Alias-Dajczer 5, Alias-Bessa-Dajczer 1 and Alias-Impera-Rigoli 6.
