The solution of the graphon mean field game equation (GMFG) is characterized by an HJB equation and an FPK equation coupled via a graphon mean field. An analysis of the classical solution of the GMFG equation system on Holder spaces is provided. Within this framework, a solution to the best response control problem is derived yielding an operator which regenerates the graphon mean field term. This operator is shown to be a Lipschitz mapping which is contractive under specified conditions which consequently yield the existence and uniqueness of the solution to the GMFG equation system.