Homogenization of nonlinear elliptic systems in nonreflexive Musielak-Orlicz spaces

We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic $N-$function, which may be possibly also dependent on the spatial variable, i.e., the homogenization process will change the characteristic function spaces at each step. Such a problem is well known and there exists many positive results for the function satisfying ${\mathrm{\Delta }}_2$ and ${\mathrm{\nabla }}_2$ conditions an being in addition H"{o}lder continuous with respect to the spatial variable. We shall show that cases these conditions can be neglected and will deal with a rather general problem in general function space setting.