A homogenization problem is considered for the periodic elliptic differential operators on $ L_2(\Pi )$, $ \Pi =\mathbb{R} \times (0, a)$, defined by the differential expression $\displaystyle \mathcal {B}_{\lambda }^{\varepsilon } = \sum _{j=1}^2 \mathrm {D... ...repsilon , x_2)\mathrm {D}_j + \mathrm {D}_j h_j(x_1/\varepsilon , x_2) \bigr )$ $\displaystyle + \ Q(x_1/\varepsilon , x_2) + \lambda Q_*(x_1/\varepsilon x_2)$ with periodic, Neumann, or Dirichlet boundary conditions. The coefficients of the expression are assumed to be periodic of period $ 1$ in the first variable and smooth in some sense in the second. Sharp-order approximations are obtained for the inverse of $ \mathcal {B}_{\lambda }^{\varepsilon }$ with respect to $ \mathbf {B}\bigl ( L_2(\Pi )\bigr )$- and $ \mathbf {B}\bigl ( L_2(\Pi ), H^1(\Pi ) \bigr )$-norms in the small $ \varepsilon $ limit with error terms of order $ \varepsilon $.