  We give a new perspective on the existence of viscosity solutions for a
stationary and a time-dependent first order Hamilton-Jacobi equation. Following
recent comparison principles, we work in a framework in which we consider a
subsolution and a supersolution for two equations in terms of two Hamiltonians
that can be seen as an upper and lower bound of our original Hamiltonian
respectively. The upper and lower bound are formulated in terms of a relaxed
Lyapunov function which allows us to restrict part of the analysis to compact
sets and to work with almost optimizers of the considered control problems. For
this reason, we can relax assumptions on the control problem: most notably, we
do not need completeness of set of controlled paths. Moreover, our strategy
avoids a-priori analysis on the regularity of the candidate solutions. We then
explore the context of Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs
equations to verify our assumptions in a convex and non-convex setting
respectively.
