  Previous results about the non trivial solution of the $\Phi^4$-equations of
motion for the Green's functions in the Euclidean\ space (of $0\leq r\leq 4$
dimensions) in the Wightman Quantum Field theory framework, are reviewed in the
0 - dimensional case from the following three aspects: The structure of the
subset $\Phi \subset {\cal B}$ characterized by the signs and "splitting"
(factorization properties) is reffined and more explictly described by a subset
$\Phi_0\subset \Phi$
  The local contractivity of the corresponding $\Phi ^4_0$ mappping is
established in a neighborhood of a precise nontrivial sequence $H_0\in \Phi_0$
using a new norm in the Banach space $ {\cal B}$. A new $\Phi ^4_0$ iteration
is defined in the neighborhood of this sequence $H_0\in \Phi_0 $ and our
numerical study displays clearly the stability of $\Phi_0$ (the splitting, the
bounds and sign properties are perfectly illustrated), and a rapid convergence
to the unique fixed point $H^* \in \Phi_0$.
