  We show the existence and uniqueness of a solution to a $\Phi_4^4$ non linear
renormalized system of equations of motion in Euclidean space. This system
represents a non trivial model which describes the dynamics of the $\Phi_4^4$
Green's functions in the Axiomatic Quantum Field Theory (AQFT) framework.
  The main argument is the local contractivity of the so called \emph{"new
mapping"} in the neighborhood of a particular "tree type" sequence of Green's
functions. This neighborhood (and the $\Phi_4^4$ non trivial solution) belongs
to a particular subset of the appropriate Banach space characterized by signs,
splitting (analogous to that of the $\Phi_0^4$ solution), axiomatic analyticity
properties and "good" asymptotic behavior with respect to the four-dimensional
euclidean external momenta.
