  We investigate two inertial forward-backward algorithms in connection with
the minimization of the sum of a non-smooth and possibly non-convex and a
non-convex differentiable function. The algorithms are formulated in the spirit
of the famous FISTA method, however the setting is non-convex and we allow
different inertial terms. Moreover, the inertial parameters in our algorithms
can take negative values too. We also treat the case when the non-smooth
function is convex and we show that in this case a better step size can be
allowed. We prove some abstract convergence results which applied to our
numerical schemes allow us to show that the generated sequences converge to a
critical point of the objective function, provided a regularization of the
objective function satisfies the Kurdyka-Lojasiewicz property. Further, we
obtain a general result that applied to our numerical schemes ensures
convergence rates for the generated sequences and for the objective function
values formulated in terms of the KL exponent of a regularization of the
objective function. Finally, we apply our results to image restoration.
