FRW in cosmological self-creation theory

We use the Brans-Dicke theory from the framework of General Relativity (Einstein frame), but now the total energy momentum tensor fulfills the following condition $\rm <[{1}{\phi}T^{\mu \nu M}+T^{\mu \nu}(\phi)>]_{;\nu}=0$. We take as a first model the flat FRW metric and with the law of variation for Hubble’s parameter proposal by Berman \cite{Berman}, we find solutions to the Einstein field equations by the cases: inflation ($\gamma =-1$), radiation (\gamma=\{1}{3}), stiff matter ($\gamma =1$). For the Inflation case the scalar field grows fast and depends strongly of the constant ${\mathrm{M}}_{\gamma =-1}$ that appears in the solution, for the Radiation case, the scalar stop its expansion and then decrease perhaps due to the presence of the first particles. In the Stiff Matter case, the scalar field is decreasing so for a large time, $\varphi \to 0$. In the same line of classical solutions, we find an exact solution to the Einstein field equations for the stiff matter $(\gamma =1)$ and flat universe, using the Hamilton-Jacobi scheme.