  In this paper we analyse a boundary value problem for the Laplace equation
with a nonlinear non-autonomous transmission conditions on the boundary of a
small inclusion of size $\epsilon$. We show that the problem has solutions for
$\epsilon$ small enough and we investigate the dependence of a specific family
of solutions upon $\epsilon$. By adopting a functional analytic approach we
prove that the map which takes $\epsilon$ to (suitable restrictions of) the
corresponding solution can be represented in terms of real analytic functions.
