Geodesic
Convergence theorems for asymptotically nonexpansive mappings in Banach spaces
Acta mathematica Universitatis Comenianae, Tome 77 (2008) no. 1 
Yongfu Su; Xiaolong Qin; Meijuan Shang. Convergence theorems  for  asymptotically nonexpansive mappings in Banach spaces. Acta mathematica Universitatis Comenianae, Tome 77 (2008) no. 1. http://geodesic.mathdoc.fr/item/AMUC_2008_77_1_a3/
@article{AMUC_2008_77_1_a3,
     author = {Yongfu Su and Xiaolong Qin and Meijuan Shang},
     title = {Convergence theorems  for  asymptotically nonexpansive mappings in {Banach} spaces},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2008},
     volume = {77},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2008_77_1_a3/}
}
TY  - JOUR
AU  - Yongfu Su
AU  - Xiaolong Qin
AU  - Meijuan Shang
TI  - Convergence theorems  for  asymptotically nonexpansive mappings in Banach spaces
JO  - Acta mathematica Universitatis Comenianae
PY  - 2008
VL  - 77
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/AMUC_2008_77_1_a3/
ID  - AMUC_2008_77_1_a3
ER  - 
%0 Journal Article
%A Yongfu Su
%A Xiaolong Qin
%A Meijuan Shang
%T Convergence theorems  for  asymptotically nonexpansive mappings in Banach spaces
%J Acta mathematica Universitatis Comenianae
%D 2008
%V 77
%N 1
%U http://geodesic.mathdoc.fr/item/AMUC_2008_77_1_a3/
%F AMUC_2008_77_1_a3

Voir la notice de l'article provenant de la source Comenius University

Let E be a uniformly convex Banach space, and let K be a nonempty convex closed subset which is also a nonexpansive retract of E . Let T : K ® E be an asymptotically nonexpansive mapping with { kn } Ì [1, ¥ ) such that ( å from n =1 to ¥ ) ( kn - 1) < ¥ and let F ( T ) be nonempty, where F ( T ) denotes the fixed points set of T . Let {a n } , {b n } , {g n } , {a¢ n } , {b¢ n } , {g¢ n } , {a¢¢ n } , {b¢¢ n } and {g¢¢ n } be real sequences in [0, 1] such that a n + b n + g n = a¢ n + b¢ n + g¢ n = a¢¢ n + b¢¢ n + g¢¢ n = 1 and e £ a n , a¢ n , a¢¢ n £ 1 - e for all n Î N and some e > 0 , starting with arbitrary x 1 Î K , define the sequence { xn } by setting zn = P ( a¢¢ n T ( PT ) n-1 xn + b¢¢ n xn + g¢¢ n wn ), yn = P ( a¢ n T ( PT ) n-1 zn + b¢ n xn + g¢ n vn ), x n+1 = P ( a n T ( PT ) n-1 yn + b n xn + g n un ), with the restrictions ( å from n =1 to ¥ ) ( g n ) < ¥ , ( å from n =1 to ¥ ) ( g¢ n ) < ¥ and ( å from n =1 to ¥ ) ( g¢¢ n ) < ¥ where { wn } , { vn } and { un } are bounded sequences in K . (i) If E is real uniformly convex Banach space satisfying Opial's condition, then weak convergence of { xn } to some p Î F ( T ) is obtained; (ii) If T satisfies condition (A), then { xn } convergence strongly to some p Î F ( T ).