×
Sitthikul, Kittipong; Saejung, Satit

Convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings. (English) Zbl 1192.47063

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 48, 139-152 (2009).
Summary: Weak and strong convergence of finite step iteration sequences to a common fixed point for a pair of a finite family of nonexpansive mappings and a finite family of asymptotically nonexpansive mappings in a nonempty closed convex subset of uniformly convex Banach spaces are presented.

Cited in 2 Reviews
Cited in 3 Documents

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems

Keywords:

weak convergence; asymptotically nonexpansive mapping; common fixed point; finite-step iterative sequence
Cite Review PDF
Full Text: EuDML

References:

[1] Chidume, C. E., Ali, B.: Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 326, 2 (2007), 960-973. · Zbl 1112.47053 · doi:10.1016/j.jmaa.2006.03.045
[2] Chidume, C. E., Ali, B.: Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 330, 1 (2007), 377-387. · Zbl 1141.47039 · doi:10.1016/j.jmaa.2006.07.060
[3] Cho, Y. J., Zhou, H., Guo, G.: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 47, 4-5 (2004), 707-717. · Zbl 1081.47063 · doi:10.1016/S0898-1221(04)90058-2
[4] Clarkson, J. A.: Uniformly convex spaces. Trans. Amer. Math. Soc. 40, 3 (1936), 396-414. · Zbl 0015.35604 · doi:10.2307/1989630
[5] García-Falset, J., Kaczor, W., Kuczumow, T., Reich, S.: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., Ser. A: Theory Methods 43, 3 (2001), 377-401. · Zbl 0983.47040 · doi:10.1016/S0362-546X(99)00200-X
[6] Goebel, K., Kirk, W. A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. 35 (1972), 171-174. · Zbl 0256.47045 · doi:10.2307/2038462
[7] Gornicki, J.: Weak and strong convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. Math. Univ. Carolin 30 (1989), 249-252.
[8] Ishikawa, S.: Fixed points by new iteration method. Proc. Amer. Math. Soc. 44 (1974), 147-150. · Zbl 0286.47036 · doi:10.2307/2039245
[9] Liu, Z., Agarwal, R. P., Feng, C., Kang, S. M.: Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 44 (2005), 83-96. · Zbl 1099.47057
[10] Liu, Z., Feng, C., Ume, J. S., Kang, S. M.: Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings. Taiwanese J. Math. 11, 1 (2007), 27-42. · Zbl 1151.47060
[11] Mann, W. R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4 (1953), 506-510. · Zbl 0050.11603 · doi:10.2307/2032162
[12] Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967), 591-597. · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[13] Osilike, M. O., Aniagbosor, S. C.: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Comput. Modelling 32, 10 (2000), 1181-1191. · Zbl 0971.47038 · doi:10.1016/S0895-7177(00)00199-0
[14] Schu, J.: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158, 2 (1991), 407-413. · Zbl 0734.47036 · doi:10.1016/0022-247X(91)90245-U
[15] Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. 43, 1 (1991), 153-159. · Zbl 0709.47051 · doi:10.1017/S0004972700028884
[16] Senter, H. F., Dotson, W. G., Jr: Approximating fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. 44 (1974), 375-380. · Zbl 0299.47032 · doi:10.2307/2040440
[17] Tan, K. K., Xu, H. K.: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. 122 (1994), 733-739. · Zbl 0820.47071 · doi:10.2307/2160748
[18] Thakur, B. S., Jung, J. S.: Weak and strong convergence of multistep iteration for finite family of asymptotically nonexpansive mappings. Fixed Point Theory Appl., Art. ID 31056 (2007), 15 pp. · Zbl 1155.47315 · doi:10.1155/2007/31056
[19] Xu, B., Noor, M. A.: Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 267 (2002), 444-453. · Zbl 1011.47039 · doi:10.1006/jmaa.2001.7649
[20] Zhou, Y. Y., Chang, S. S.: Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numer. Funct. Anal. and Optimiz. 23 (2002), 911-921. · Zbl 1041.47048 · doi:10.1081/NFA-120016276
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.