  The critical temperature of high-$T_c$ superconductors is determined, at
least in part, by the electron-phonon coupling. We include the effect of an
exchange interaction between the electrons and calculate the renormalization of
the bare phonon frequencies and the electron-phonon verticies in a random phase
approximation and obtain a strongly enhanced attractive phonon-induced
electron-electron interaction. Using Fast Fourier Transform techniques, the
weak-coupling selfconsistency equation for the order parameter is solved in the
2D first Brillouin zone for the Emery tight-binding band with different band
fillings. The enhancement of $T_c$ arises primarily from the softening of the
phonon frequencies rather than the vertex renormalization.
