Copyright © 2010 Shuang Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In the paper by Hu in 2008, the author proved a strong convergence result for nonexpansive mappings using a modified Halpern's iteration algorithm. Unfortunately, the case limn→∞βn=1 does not guarantee the strong convergence of the sequence {xn}. In this note, we provide a counter-example to the theorem.
In [1], the author introduced a modified Halpern’s iteration. For any 
, the sequence is defined by
here , and are three real sequences in , satisfying . The author proved the following strong convergence theorem.
Theorem 1 (see [1]).
Let
be a nonempty closed convex subset of a real Banach space
which has a uniformly Gâteaux differentiable norm. Let
be a nonexpansive mapping with
. Assume that
converges strongly to a fixed point
of
as
, where
is the unique element of
which satisfies
for any
. Let
, and
be three real sequences in
which satisfy the following conditions:
and
. For any
, the sequence
is defined by the iteration in (). Then the sequence
converges strongly to a fixed point of
.
Counter Example
Let be a real Banach space whose norm is uniformly Gâteaux differentiable. Let be a nonempty closed and convex subset of , defined by
(1) where , with a fixed element of . Let be a mapping defined by for all . It is obvious that is a nonexpansive mapping and . Take , and for all and . We also can obtain that . Observe that all conditions of Theorem 1 are satisfied. However, the iterative sequence does not converge strongly to the fixed point of .
Claim 1.
If , then .
Proof.
In fact, we have
(2)where can be denoted as . If , then . From the above equality we have
(3)Hence does not converge strongly to .
Remark 1.
Why does the proof of Theorem 1 fail? It is not difficult to check that the proof of Case 2 () is not suitable.
Acknowledgments
This work is supported by the National Science Foundation of China under Grant no. 10771175 and the Natural Science Foundational Committee of Hubei Province (D200722002).
References
- L.-G. Hu, “Strong convergence of a modified Halpern's iteration for nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 649162, 9 pages, 2008.