  We derive a direct connection between Nesterov's accelerated first-order
algorithm and the Halpern fixed-point iteration scheme for approximating a
solution of a co-coercive equation. We show that one method is equivalent to
another via a simple transformation, leading to a simple convergence proof for
Nesterov's accelerated scheme. Alternatively, we directly establish convergence
rates of Nesterov's accelerated variant, and as a consequence, we obtain a new
convergence rate of the Halpern fixed-point iteration. Next, we apply our
results to different methods such as proximal-point, forwarb-backward
splitting, and three-operator splitting methods for solving monotone
inclusions, where our convergence guarantees are applicable. Since the forward
scheme requires the co-coerciveness of the underlying operator, we derive new
Nesterov's accelerated variants for both recent extra-anchored gradient and
past-extra anchored gradient methods which can avoid the co-coerciveness
assumption. For the extra-anchored gradient method, we consider two variants
studied in [43] for the monotone case and in [21] for the co-monotone case. The
latter covers the former as a special case and has a tighter convergence rate.
For the past-extra anchored gradient method (an accelerated variant of the
optimistic gradient method) [41], we obtain a new Nesterov's accelerated
variant with two past-iterate correction terms that may help to develop new
Nesterov's accelerated methods for minimax problems and their continuous views.
We test our theoretical results on two numerical examples, where the actual
convergence rates match well the theoretical rates up to a constant factor.
