Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations

For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^*Mtimesmathbb R$ which is conex in $p$ and monotonically increasing in $u$. Let $u_epsilon^-:Mrightarrowmathbb R$ be the viscosity solution of the parametrized contact Hamilton-Jacobi equation [ H(x,partial_x u_epsilon^-(x),epsilon u_epsilon^-(x))=c(H) ] with $c(H)$ being the Ma~ne Critical Value. We prove that $u_epsilon^-$ converges uniformly, as $epsilonrightarrow 0_+$, to a specfic viscosity solution $u_0^-$ of the critical equation [ H(x,partial_x u_0^-(x),0)=c(H) ] which can be characterized as a minimal combination of associated Peierls barrier functions.