Strong convergence and control condition of modified Halpern iterations in Banach spaces

Yonghong Yao, Rudong Chen and Haiyun Zhou

International Journal of Mathematics and Mathematical Sciences, 2006, vol. 2006, 1-10

Abstract:

Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gâteaux differentiable norm. Let T ∈ Γ C and f ∈ Π C . Assume that { x t } converges strongly to a fixed point z of T as t → 0 , where x t is the unique element of C which satisfies x t = t f ( x t ) + ( 1 − t ) T x t . Let { α n } and { β n } be two real sequences in ( 0 , 1 ) which satisfy the following conditions: ( C 1 ) lim ⁡ n → ∞ α n = 0 ; ( C 2 ) ∑ n = 0 ∞ α n = ∞ ; ( C 6 ) 0 < lim ⁡ inf ⁡ n → ∞ β n ≤ lim ⁡ sup ⁡ n → ∞ β n < 1 . For arbitrary x 0 ∈ C , let the sequence { x n } be defined iteratively by y n = α n f ( x n ) + ( 1 − α n ) T x n , n ≥ 0 , x n + 1 = β n x n + ( 1 − β n ) y n , n ≥ 0 . Then { x n } converges strongly to a fixed point of T .

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:029728

DOI: 10.1155/IJMMS/2006/29728

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