We consider a general second order matrix operator in a multi-dimensional
domain subject to a classical boundary condition. This operator is perturbed by
a first order differential operator, the coefficients of which depend
arbitrarily on a small multi-dimensional parameter. We study the existence of a
limiting (homogenized) operator in the sense of the norm resolvent convergence
for such perturbed operator. The first part of our main results states that the
norm resolvent convergence is equivalent to the convergence of the coefficients
in the perturbing operator in certain space of multipliers. If this is the
case, the resolvent of the perturbed operator possesses a complete asymptotic
expansion, which converges uniformly to the resolvent. The second part of our
results says that the convergence in the mentioned spaces of multipliers is
equivalent to the convergence of certain local mean values over small pieces of
the considered domains. These results are supported by series of examples. We
also provide a series of ways of generating new non-periodically oscillating
perturbations, which finally leads to a very wide class of perturbations, for
which our results are applicable