  We construct the generalized Darboux transformation with arbitrary functions
in time $t$ for the AKNS equation with self-consistent sources (AKNSESCS)
which, in contrast with the Darboux transformation for the AKNS equation,
provides a non-auto-B\"{a}cklund transformation between two AKNSESCSs with
different degrees of sources. The formula for N-times repeated generalized
Darboux transformation is proposed. By reduction the generalized Darboux
transformation with arbitrary functions in time $t$ for the Nonlinear
Schr\"{o}dinger equation with self-consistent sources (NLSESCS) is obtained and
enables us to find the dark soliton, bright soliton and positon solutions for
NLS$^{+}$ESCS and NLS$^{-}$ESCS. The properties of these solution are analyzed.
