  Let $X$ be a smooth $n\,$-dimensional manifold and $D$ be an open connected
set in $X$ with smooth boundary $\partial D$. Perturbing the Cauchy problem for
an elliptic system $Au = f$ in $D$ with data on a closed set $\iG \subset
\partial D$ we obtain a family of mixed problems depending on a small parameter
$\varepsilon > 0$. Although the mixed problems are subject to a non-coercive
boundary condition on $\partial D \setminus \iG$ in general, each of them is
uniquely solvable in an appropriate Hilbert space $\cD_{T}$ and the
corresponding family $\{ u_{\varepsilon} \}$ of solutions approximates the
solution of the Cauchy problem in $\cD_{T}$ whenever the solution exists. We
also prove that the existence of a solution to the Cauchy problem in $\cD_{T}$
is equivalent to the boundedness of the family $\{ u_{\varepsilon} \}$. We thus
derive a solvability condition for the Cauchy problem and an effective method
of constructing its solution. Examples for Dirac operators in the Euclidean
space $\R^n$ are considered. In the latter case we obtain a family of mixed
boundary problems for the Helmholtz equation.
