# Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

Open Mathematics (2007)

• Volume: 5, Issue: 2, page 345-357
• ISSN: 2391-5455

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Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T 1, T 2 and T 3: K → E be asymptotically nonexpansive mappings with k n, l n and j n. [1, ∞) such that Σn=1∞(k n − 1) < ∞, Σn=1∞(l n − 1) < ∞ and Σn=1∞(j n − 1) < ∞, respectively and F nonempty, where F = x ∈ K: T 1x = T 2x = T 3 x = xdenotes the common fixed points set of T 1, T 2 and T 3. Let α n, α′ n and α″ n be real sequences in (0, 1) and ∈ ≤ α n, α′ n, α″ n ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x 1 ∈ K define the sequence x n by $\left\{\begin{array}{c}{z}_{n}=P\left({\alpha }_{n}^{\text{'}\text{'}}{T}_{3}{\left(P{T}_{3}\right)}^{n-1}{x}_{n}+\left(1-{\alpha }_{n}^{\text{'}\text{'}}\right){x}_{n}\right),\\ {y}_{n}=P\left({\alpha }_{n}^{\text{'}}{T}_{2}{\left(P{T}_{2}\right)}^{n-1}{z}_{n}+\left(1-{\alpha }_{n}^{\text{'}}\right){x}_{n}\right),\\ {x}_{n+1}=P\left({\alpha }_{n}{T}_{1}{\left(P{T}_{1}\right)}^{n-1}{y}_{n}+\left(1-{\alpha }_{n}\right){x}_{n}\right).\end{array}\right$/extract_itex] (i) If the dual E* of E has the Kadec-Klee property then x n converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then x n converges strongly to a common fixed point p ∈ F. ## How to cite top Xiaolong Qin, Yongfu Su, and Meijuan Shang. "Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces." Open Mathematics 5.2 (2007): 345-357. <http://eudml.org/doc/269414>. @article{XiaolongQin2007, abstract = {Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T 1, T 2 and T 3: K → E be asymptotically nonexpansive mappings with k n, l n and j n. [1, ∞) such that Σn=1∞(k n − 1) < ∞, Σn=1∞(l n − 1) < ∞ and Σn=1∞(j n − 1) < ∞, respectively and F nonempty, where F = x ∈ K: T 1x = T 2x = T 3 x = xdenotes the common fixed points set of T 1, T 2 and T 3. Let α n, α′ n and α″ n be real sequences in (0, 1) and ∈ ≤ α n, α′ n, α″ n ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x 1 ∈ K define the sequence x n by \[\left\lbrace \{\begin\{array\}\{c\}z\_n = P(\alpha ^\{\prime \prime \}\_n T\_3 (PT\_3 )^\{n - 1\} x\_n + (1 - \alpha ^\{\prime \prime \}\_n )x\_n ), \hfill \\ y\_n = P(\alpha ^\{\prime \}\_n T\_2 (PT\_2 )^\{n - 1\} z\_n + (1 - \alpha ^\{\prime \}\_n )x\_n ), \hfill \\ x\_\{n + 1\} = P(\alpha \_n T\_1 (PT\_1 )^\{n - 1\} y\_n + (1 - \alpha \_n )x\_n ). \hfill \\ \end\{array\}\} \right.$ (i) If the dual E* of E has the Kadec-Klee property then x n converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then x n converges strongly to a common fixed point p ∈ F. },
author = {Xiaolong Qin, Yongfu Su, Meijuan Shang},
journal = {Open Mathematics},
keywords = {Asymptotically nonexpansive; non-self map; Kadec-Klee property; Uniformly convex; asymptotically nonexpansive mapping; uniformly convex spaces},
language = {eng},
number = {2},
pages = {345-357},
title = {Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces},
url = {http://eudml.org/doc/269414},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Xiaolong Qin
AU - Yongfu Su
AU - Meijuan Shang
TI - Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces
JO - Open Mathematics
PY - 2007
VL - 5
IS - 2
SP - 345
EP - 357
AB - Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T 1, T 2 and T 3: K → E be asymptotically nonexpansive mappings with k n, l n and j n. [1, ∞) such that Σn=1∞(k n − 1) < ∞, Σn=1∞(l n − 1) < ∞ and Σn=1∞(j n − 1) < ∞, respectively and F nonempty, where F = x ∈ K: T 1x = T 2x = T 3 x = xdenotes the common fixed points set of T 1, T 2 and T 3. Let α n, α′ n and α″ n be real sequences in (0, 1) and ∈ ≤ α n, α′ n, α″ n ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x 1 ∈ K define the sequence x n by $\left\lbrace {\begin{array}{c}z_n = P(\alpha ^{\prime \prime }_n T_3 (PT_3 )^{n - 1} x_n + (1 - \alpha ^{\prime \prime }_n )x_n ), \hfill \\ y_n = P(\alpha ^{\prime }_n T_2 (PT_2 )^{n - 1} z_n + (1 - \alpha ^{\prime }_n )x_n ), \hfill \\ x_{n + 1} = P(\alpha _n T_1 (PT_1 )^{n - 1} y_n + (1 - \alpha _n )x_n ). \hfill \\ \end{array}} \right.$ (i) If the dual E* of E has the Kadec-Klee property then x n converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then x n converges strongly to a common fixed point p ∈ F.
LA - eng
KW - Asymptotically nonexpansive; non-self map; Kadec-Klee property; Uniformly convex; asymptotically nonexpansive mapping; uniformly convex spaces
UR - http://eudml.org/doc/269414
ER -

## References

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