Vanishing contact structure problem and convergence of the viscosity solutions

This article is devoted to studying the vanishing contact structure problem which is a generalization of the vanishing discount problem. Let be a family of Hamiltonians of contact type with parameter and converge to G(x, p). For the contact type Hamilton-Jacobi equation with respect to we prove that, under mild assumptions, the associated viscosity solution 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.