  We extend the Barles-Perthame procedure of semi-relaxed limits of viscosity
solutions of Hamilton-Jacobi equations of the type f - lambda H f = h.
  The convergence result allows for equations on a `converging sequence of
spaces' as well as Hamilton-equations written in terms of two equations in
terms of operators H_\dagger and H_\dagger that serve as natural upper and
lower bounds for the `true' operator H.
  In the process, we establish a strong relation between non-linear
pseudo-resolvents and viscosity solutions of Hamilton-Jacobi equations. As a
consequence we derive a convergence result for non-linear semigroups.
