= Solution
To first order in $n_\epsilon$, replace $q$ by $2n_\epsilon$ and use the known incident field in the outgoing integral. The <Born approximation for scalar wave scattering> gives
$$
\boxed{\psi_s(\mathbf r)\simeq2k_0^2\int_BG_{k_0}(\mathbf r,\mathbf r')\psi_{inc}(\mathbf r')n_\epsilon(\mathbf r')\,d\mathbf r'.}
$$
Both the $n_\epsilon^2$ contrast term and the internal-field correction are second order for fixed geometry in the perturbative regime established in part i.
Specify the data space before taking an <adjoint operator>. For example, let $X=L^2(B)$ and $Y=L^2(\Gamma)$, where $\Gamma$ is a bounded measurement surface separated from $B$. Define the <linear operator>
$$
(Af)(\mathbf r)=2k_0^2\int_BG_{k_0}(\mathbf r,\mathbf r')\psi_{inc}(\mathbf r')f(\mathbf r')\,d\mathbf r',\qquad\mathbf r\in\Gamma.
$$
For bounded incident field this is a <Hilbert-Schmidt operator>, hence a <compact operator>. The equation is $An_\epsilon=y$, with $y=\psi_s|_\Gamma$. Other sampling geometries give corresponding data spaces and weights; the paper does not specify one. Using the usual complex $L^2$ inner products, its <adjoint operator> is
$$
(A^*y)(\mathbf r')=2k_0^2\overline{\psi_{inc}(\mathbf r')}\int_\Gamma\overline{G_{k_0}(\mathbf r,\mathbf r')}y(\mathbf r)\,dS(\mathbf r).
$$
The complex conjugations are required by the <adjoint operator> identity, not by <wave reciprocity> alone.
The <Landweber iteration> starts from $f_0=0$ and applies <gradient descent> to $\frac12\|Af-y\|^2$:
$$
\boxed{f_{m+1}=f_m+\tau A^*(y-Af_m),\qquad0<\tau<\frac2{\|A\|^2}.}
$$
Each step back-propagates the data residual. The <Landweber relaxation parameter> controls stability, and <early stopping of Landweber iteration> prevents small <singular values> from amplifying <measurement errors>. For an explicitly real-valued index contrast, use the real <Hilbert space> structure and replace $A^*$ by $\operatorname{Re}A^*$ in this update. Additional sign or support constraints require corresponding projections; none are assumed here.
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