  This is a sequel to the paper [Oh5] (or ArXiv:math.SG/0206092). The main
purpose of the paper is to give the proof of an existence theorem, with energy
bounds, of certain pseudo-holomorphic sections of the mapping cylinder that is
needed for the proof of nondegeneracy of the homological invariant pseudo-norm
which the author has constructed on general symplectic manifolds [Oh4,5]. The
existence theorem is also the crux of the author's recent proof of an optimal
energy-capacity inequality given in [Oh5]. In this paper, we prove a more
general existence result than needed in that we study Floer's perturbed
Cauchy-Riemann equations with discontinous Hamiltonian perturbation terms and
prove an existence theorem of certain piecewise smooth finite energy solutions
of the equation. The proof relies on a careful study of the product structure
in the chain level Floer homology theory and a singular degeneration
(``adiabatic degeneration'') of Floer's perturbed Cauchy-Riemann equation. In
the course of the proof, we also derive certain general energy identity of
pseudo-holomorphic sections of the Hamiltonian fibration.
