  This paper is concerned with the study of Aubry-Mather and weak KAM theories
for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on
$T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and
$0<\frac{\partial H}{\partial u}\leqslant \lambda$ for some $\lambda>0$, where
$M$ is a connected, closed and smooth manifold. First, we show the uniqueness
of the backward weak KAM solutions of the corresponding Hamilton-Jacobi
equation. Using the unique backward weak KAM solution $u_-$, we prove the
existence of the maximal forward weak KAM solution $u_+$. Next, we analyse
Aubry set for the contact Hamiltonian system showing that it is the
intersection of two Legendrian pseudographs $G_{u_-}$ and $G_{u_+}$, and that
the projection $\pi:T^*M\times \mathbb{R}\to M$ induces a bi-Lipschitz
homeomorphism $\pi|_{\tilde{\mathcal{A}}}$ from Aubry set $\tilde{\mathcal{A}}$
onto the projected Aubry set $\mathcal{A}$. At last, we introduce the notion of
barrier functions and study their interesting properties along calibrated
curves. Our analysis is based on a recent method by [43,44].
