  In this paper, we generalize weak KAM theorem from positive Lagrangian
systems to "proper" Hamilton-Jacobi equations.
  We introduce an implicitly defined solution semigroup of evolutionary
Hamilton-Jacobi equations. By exploring the properties of the solution
semigroup, we prove the convergence of solution semigroup and existence of weak
KAM solutions for stationary equations: \begin{equation*} H(x, u, d_x u)=0.
\end{equation*}
