  The "Inertial Forward-Backward algorithm" (IFB) is a powerful tool for convex
nonsmooth minimization problems, it gives the well known "fast iterative
shrinkage-thresholding algorithm " (FISTA), which enjoys $O\left(
{\frac{1}{{{k^2}}}} \right)$ global convergence rate of function values,
however, no convergence of iterates has been proved; by do a small
modification, an accelerated IFB called "FISTA\_CD" improves the convergence
rate of function values to $o\left( {\frac{1}{{{k^2}}}} \right)$ and shows the
weak convergence of iterates. The local error bound condition is extremely
useful in analyzing the convergence rates of a host of iterative methods for
solving optimization problems, and in practical application, a large number of
problems with special structure often satisfy the error bound condition.
Naturally, using local error bound condition to derive or improve the
convergence rate of IFB is a common means. In this paper, based on the local
error bound condition, we exploit an new assumption condition for the important
parameter $t_k$ in IFB, and establish the improved convergence rate of function
values and strong convergence of the iterates generated by the IFB algorithms
with six $t_k$ satisfying the above assumption condition in Hilbert space. It
is remarkable that, under the local error bound condition, we establish the
strong convergence of the iterates generated by the original FISTA, and prove
that the convergence rates of function values for FISTA\_CD is actually related
to the value of parameter $a,$ and show that the IFB algorithms with some $t_k$
mentioned above can achieve sublinear convergence rate $o\left(
{\frac{1}{{{k^p}}}} \right)$ for any positive integer $p>1$. Some numerical
experiments are conducted to illustrate our results.
