On Favardʹs theorems

In this paper, we improve and extend the classical Favardʹs theorems, i.e. Favardʹs theorem of the module containment, Favardʹs theorem of linear differential equations. We study Favardʹs theory of linear differential equations with piecewise constant argument. An example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument. The equivalence between almost automorphic functions and N-almost periodic ones is studied.