Vanishing contact structure problem and convergence of the viscosity solutions

This paper is devoted to study the vanishing contact structure problem which is a generalization of the vanishing discount problem. Let $H^\lambda(x,p,u)$ be a family of Hamiltonians of contact type with parameter $\lambda>0$ and converges to $G(x,p)$. For the contact type Hamilton-Jacobi equation with respect to $H^\lambda$, we prove that, under mild assumptions, the associated viscosity solution $u^{\lambda}$ converges to a specific viscosity solution $u^0$ of the vanished contact equation. As applications, we give some convergence results for the nonlinear vanishing discount problem.