  Theories that support dynamical generation of a fermion mass gap are of
widespread interest. The phenomenon is often studied via the Dyson-Schwinger
equation (DSE) for the fermion self energy; i.e., the gap equation. When the
rainbow truncation of that equation supports dynamical mass generation, it
typically also possesses a countable infinity of simultaneous solutions for the
dressed-fermion mass function, solutions which may be ordered by the number of
zeros they exhibit. These features can be understood via the theory of
nonlinear Hammerstein integral equations. Using QED3 as an example, we
demonstrate the existence of a large class of gap equation truncations that
possess solutions with damped oscillations. We suggest that there is a larger
class, quite probably including the exact theory, which does not. The structure
of the dressed-fermion--gauge-boson vertex is an important factor in deciding
the issue.
