p. 31 - 42 Convergence theorems for asymptotically nonexpansive mappings in Banach spaces Yongfu Su, Xiaolong Qin and Meijuan Shang Received: September 09, 2006; Accepted: May 22, 2007 Abstract. 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 {an}, {bn}, {gn}, {a¢n}, {b¢n}, {g¢n}, {a¢¢n}, {b¢¢n} and {g¢¢n} be real sequences in [0, 1] such that an + bn + gn = a¢n + b¢n + g¢n = a¢¢n + b¢¢n + g¢¢n = 1 and e £ an, a¢n, a¢¢n £ 1 - e for all n Î N and some e > 0, starting with arbitrary x1 Î K, define the sequence { xn} by setting zn = P(a¢¢nT(PT)n-1xn + b¢¢nxn + g¢¢nwn), yn = P(a¢nT(PT)n-1zn + b¢nxn + g¢nvn), xn+1 = P(anT(PT)n-1yn + bnxn + gnun), with the restrictions (å from n=1 to ¥) (gn) ¥, (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). Keywords: asymptotically nonexpansive; non-self map; composite iterative with errors; Kadec-Klee property; Uniformly convex Banach space. AMS Subject classification: Primary: 47H09; 47H10; 47J25. PDF Compressed Postscript Version to read ISSN 0862-9544 (Printed edition) Faculty of Mathematics, Physics and Informatics Comenius University 842 48 Bratislava, Slovak Republic Telephone: + 421-2-60295111 Fax: + 421-2-65425882 e-Mail: amuc@fmph.uniba.sk Internet: www.iam.fmph.uniba.sk/amuc © 2008, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE |