We consider a contact Hamiltonian H(x, p, u) with certain dependence on the contact variable u. If \(u_{-}\) is a viscosity solution of the contact Hamilton-Jacobi equation \(\begin{aligned} H(x,D_{x}u(x),u(x))=0,\quad x\in M, \end{aligned}\) and \(u_{-}\) is locally asymptotically stable, we prove that the perturbed equation \(\begin{aligned} H(x,D_{x}u(x),u(x))+\varepsilon P(x,D_{x}u(x),u(x))=0,\quad x\in M, \end{aligned}\) has a viscosity solution \(u_{-}^{\varepsilon }\) which converges uniformly to \(u_{-}\) , as the perturbation parameter \(\varepsilon \) converges to 0. Moreover, we give a condition ensuring that in a neighborhood of the viscosity solution \(u_-\) , the perturbed solution \(u_{-}^{\varepsilon }\) is unique. Furthermore, \(u_{-}^{\varepsilon }\) is still locally Lyapunov asymptotically stable.
