  This paper proposes and develops inexact proximal methods for finding
stationary points of the sum of a smooth function and a nonsmooth weakly convex
one, where an error is present in the calculation of the proximal mapping of
the nonsmooth term. A general framework for finding zeros of a continuous
mapping is derived from our previous paper on this subject to establish
convergence properties of the inexact proximal point method when the smooth
term is vanished and of the inexact proximal gradient method when the smooth
term satisfies a descent condition. The inexact proximal point method achieves
global convergence with constructive convergence rates when the Moreau envelope
of the objective function satisfies the Kurdyka-Lojasiewicz (KL) property.
Meanwhile, when the smooth term is twice continuously differentiable with a
Lipschitz continuous gradient and a differentiable approximation of the
objective function satisfies the KL property, the inexact proximal gradient
method achieves the global convergence of iterates with constructive
convergence rates.
