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

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 -