Two-parametric error estimates in homogenization of second order elliptic systems in ${\mathbb{R}}^d$ including lower order terms

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we consider a selfadjoint operator ${\mathcal{B}}_{\epsilon }$, $0<\epsilon ⩽1$, given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D}) + \sum_{j=1}^d (a_j({\mathbf x}/\varepsilon) D_j +D_j a_j({\mathbf x}/\varepsilon)^*) + Q({\mathbf x}/\varepsilon)$,where$b({\mathbf D}) = \sum_{l=1}^d b_l D_l$isthefirstorderdifferentialoperator,and$g, a_j, Q$arematrix-valuedfunctionsin${\mathbb R}^d$ periodic with respect to some lattice $\mathrm{\Gamma }$. It is assumed that $g$ is bounded and positive definite, while $a_j$ and $Q$ are, in general, unbounded. We study the generalized resolvent $({\mathcal B}_\varepsilon - \zeta Q_0({\mathbf x}/\varepsilon))^{-1}$,where$Q_0$isa$\Gamma$-periodic, bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. Approximations for the generalized resolvent in the $(L_2\to L_2)$- and $(L_2\to H^1)$-norms with two-parametric error estimates (with respect to the parameters $\epsilon$ and $\zeta$) are obtained.