  Here, we study the selection problem for the vanishing discount approximation
of non-convex, first-order Hamilton-Jacobi equations. While the selection
problem is well understood for convex Hamiltonians, the selection problem for
non-convex Hamiltonians has thus far not been studied. We begin our study by
examining a generalized discounted Hamilton-Jacobi equation. Next, using an
exponential transformation, we apply our methods to strictly quasi-convex and
to some non-convex Hamilton-Jacobi equations. Finally, we examine a non-convex
Hamiltonian with flat parts to which our results do not directly apply. In this
case, we establish the convergence by a direct approach.
