  In this paper, we consider the convergence of an abstract inexact nonconvex
and nonsmooth algorithm. We promise a pseudo sufficient descent condition and a
pseudo relative error condition, which are both related to an auxiliary
sequence, for the algorithm; and a continuity condition is assumed to hold. In
fact, a lot of classical inexact nonconvex and nonsmooth algorithms allow these
three conditions. Under a special kind of summable assumption on the auxiliary
sequence, we prove the sequence generated by the general algorithm converges to
a critical point of the objective function if being assumed Kurdyka-
Lojasiewicz property. The core of the proofs lies in building a new Lyapunov
function, whose successive difference provides a bound for the successive
difference of the points generated by the algorithm. And then, we apply our
findings to several classical nonconvex iterative algorithms and derive the
corresponding convergence results
