  We prove existence of solutions $(\phi,\lambda)$ of a family of of
Feigenbaum-like equations \label{family} \phi(x)={1+\eps \over \lambda}
\phi(\phi(\lambda x)) -\eps x +\tau(x), where $\eps$ is a small real number and
$\tau$ is analytic and small on some complex neighborhood of $(-1,1)$ and
real-valued on $\fR$. The family $(\ref{family})$ appears in the context of
period-doubling renormalization for area-preserving maps (cf. \cite{GK}).
  Our proof is a development of ideas of H. Epstein (cf \cite{Eps1},
\cite{Eps2}, \cite{Eps3}) adopted to deal with some significant complications
that arise from the presence of terms $\eps x +\tau(x)$ in the equation
$(\ref{family})$. The method relies on a construction of novel {\it a-priori}
bounds for unimodal functions which turn out to be very tight. We also obtain
good bounds on the scaling parameter $\lambda$.
  A byproduct of the method is a new proof of the existence of a
Feigenbaum-Coullet-Tresser function.
