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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| 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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| author | Yonghong Yao Rudong Chen Haiyun Zhou |
| author_facet | Yonghong Yao Rudong Chen Haiyun Zhou |
| author_sort | Yonghong Yao |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-ef3062bca3aa4b208875ee78e0d663df |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2006-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ef3062bca3aa4b208875ee78e0d663df2025-02-03T06:08:38ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252006-01-01200610.1155/IJMMS/2006/2972829728Strong convergence and control condition of modified Halpern iterations in Banach spacesYonghong Yao0Rudong Chen1Haiyun Zhou2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, ChinaDepartment of Mathematics, Tianjin Polytechnic University, Tianjin 300160, ChinaDepartment of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, ChinaLet 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.http://dx.doi.org/10.1155/IJMMS/2006/29728 |
| spellingShingle | Yonghong Yao Rudong Chen Haiyun Zhou Strong convergence and control condition of modified Halpern iterations in Banach spaces International Journal of Mathematics and Mathematical Sciences |
| title | Strong convergence and control condition of modified
Halpern iterations in Banach spaces |
| title_full | Strong convergence and control condition of modified
Halpern iterations in Banach spaces |
| title_fullStr | Strong convergence and control condition of modified
Halpern iterations in Banach spaces |
| title_full_unstemmed | Strong convergence and control condition of modified
Halpern iterations in Banach spaces |
| title_short | Strong convergence and control condition of modified
Halpern iterations in Banach spaces |
| title_sort | strong convergence and control condition of modified halpern iterations in banach spaces |
| url | http://dx.doi.org/10.1155/IJMMS/2006/29728 |
| work_keys_str_mv | AT yonghongyao strongconvergenceandcontrolconditionofmodifiedhalperniterationsinbanachspaces AT rudongchen strongconvergenceandcontrolconditionofmodifiedhalperniterationsinbanachspaces AT haiyunzhou strongconvergenceandcontrolconditionofmodifiedhalperniterationsinbanachspaces |