Local contractivity of the $\Phi_4^4$ mapping

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.