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 does not guarantee the strong convergence of the sequence . In this note, we provide a counter-example to the theorem.
Highlights
The case limn → ∞βn 1 does not guarantee the strong convergence of the sequence {xn}
Let T : C → C be a nonexpansive mapping with Fix T / ∅
It is obvious that T is a nonexpansive mapping and Fix T {0}
Summary
In the paper by Hu in 2008, the author proved a strong convergence result for nonexpansive mappings using a modified Halpern’s iteration algorithm. The case limn → ∞βn 1 does not guarantee the strong convergence of the sequence {xn}. In 1 , the author introduced a modified Halpern’s iteration. X0 ∈ C, the sequence {xn} is defined by xn 1 αnu βnxn γnT xn, n ≥ 0, I Where {αn}, {βn}, and {γn} are three real sequences in 0, 1 , satisfying αn βn γn 1.
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