We establish two nonlinear compactness theorems in Lp(0,T;B) with hypothesis on time translations, which are nonlinear counterparts of two results by Simon (1987) . The first theorem sharpens a result by Maitre (2003)  and is important in the study of doubly nonlinear ellipticparabolic equations. Based on this theorem, we then obtain a time translation counterpart of a result by Dubinskiǐ (1965) , which is supposed to be useful in the study of some nonlinear kinetic equations (e.g. the FENE-type beadspring chains model). © 2012 Elsevier Ltd. All rights reserved.