Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 805326,2pages doi:10.1155/2010/805326
Letter to the Editor
A Note on Strong Convergence of a Modified Halpern’s Iteration for Nonexpansive Mappings
Shuang Wang
Department of Mathematics, Hubei Normal University, Huangshi 435002, China
Correspondence should be addressed to Shuang Wang,[email protected] Received 25 September 2009; Accepted 18 January 2010
Copyrightq2010 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.
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.
In1, the author introduced a modified Halpern’s iteration. For anyu, x0∈C, the sequence {xn}is defined by
xn1αnuβnxnγnTxn, n≥0, I
where{αn},{βn}, and{γn}are three real sequences in0,1, satisfyingαnβnγn 1. The author proved the following strong convergence theorem.
Theorem 1see1. LetCbe a nonempty closed convex subset of a real Banach spaceEwhich has a uniformly Gˆateaux differentiable norm. LetT :C → Cbe a nonexpansive mapping with FixT/∅.
Assume that{zt}converges strongly to a fixed pointzofT ast → 0, whereztis the unique element ofCwhich satisfieszttu1−tTzt for anyu∈C. Let{αn},{βn}, and{γn}be three real sequences in0,1which satisfy the following conditions:C1limn→ ∞αn 0 andC2∞
n0αn ∞. For anyx0 ∈C, the sequence{xn}is defined by the iteration inI. Then the sequence{xn}converges strongly to a fixed point ofT.
Counter Example
Let Ebe a real Banach space whose norm is uniformly Gˆateaux differentiable. Let Cbe a nonempty closed and convex subset ofE, defined by
2 Fixed Point Theory and Applications C
x∈E:xλy, λ∈0,3
, 1
wherey /0, withy 1 a fixed element ofE. Let T : C → Cbe a mapping defined by Tx 0 for allx∈C. It is obvious thatT is a nonexpansive mapping and FixT {0}. Take αn1/n2,βn1−2/n2, andγn1/n2for alln≥0 andx0y,u3y. We also can obtain thatzt3ty → 0t → 0. Observe that all conditions ofTheorem 1are satisfied.
However, the iterative sequence{xn}does not converge strongly to the fixed pointz0 ofT. Claim 1. Ifxn ≤1, thenxn1>xn.
Proof. In fact, we have
xn1 1 n23y
1− 2
n2
xn 1 n2Txn 3
n2y
1− 2 n2
xn
3 n2y
1− 2
n2
λny,
2
wherexn can be denoted asxn λny. Ifxn ≤ 1, then 0 < λn xn ≤ 1. From the above equality we have
xn1
3 n2
1− 2
n2
λn y 3
n2
1− 2 n2
λn
2
n21−λn 1 n2 λn
> λnxn.
3
Hence{xn}does not converge strongly toz0.
Remark 1. Why does the proof ofTheorem 1fail? It is not difficult to check that the proof of Case 2limn→ ∞βn1is 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 ProvinceD200722002.
References
1 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.
