Strong convergence and control condition of modified Halpern iterations in Banach spaces
Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gâteaux differentiable norm. Let T∈ΓC and f∈ΠC. Assume that {xt} converges strongly to a fixed point z of T as t→0, where xt is the unique element of C which satisfies xt=tf(xt)+(1−t)Txt. Let {αn} and {β...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2006-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS/2006/29728 |
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| Summary: | Let C
be a nonempty closed convex subset of a real Banach space
X
which has a uniformly Gâteaux differentiable norm. Let
T∈ΓC
and f∈ΠC. Assume that {xt}
converges
strongly to a fixed point z
of T
as t→0, where
xt
is the unique element of C
which satisfies
xt=tf(xt)+(1−t)Txt. Let {αn}
and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1)limn→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<liminfn→∞βn≤limsupn→∞βn<1. For arbitrary x0∈C, let the sequence
{xn}
be defined iteratively by
yn=αnf(xn)+(1−αn)Txn, n≥0,
xn+1=βnxn+(1−βn)yn, n≥0. Then {xn}
converges strongly to a fixed point of T. |
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| ISSN: | 0161-1712 1687-0425 |