Existence and Structure Eesults on Almost Periodic Solutions of Difference Equations.

We study the almost periodic solutions of Euler equations and of some more general difference equations. We consider two different notions of almost periodic sequences, and we establish some relations between them. We build suitable sequences spaces and we prove some properties of these spaces. We also prove properties of Nemytskii operators on these spaces. We build a variational approach to establish existence of almost periodic solutions as critical points. We obtain existence theorems for nonautonomous linear equations and for an Euler equation with a concave and coercive lagrangian. We also use a fixed point approach to obtain existence results for quasi-linear difference equations.