Previous |  Up |  Next

Article

Keywords:
Nonexpansive mapping; asymptotically nonexpansive mapping; common fixed point; finite-step iterative sequence
Summary:
In this paper, 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.
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. MR 2280956 | Zbl 1112.47053
[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. MR 2302930 | Zbl 1141.47039
[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. MR 2051342 | Zbl 1081.47063
[4] Clarkson, J. A.: Uniformly convex spaces. Trans. Amer. Math. Soc. 40, 3 (1936), 396–414. MR 1501880 | Zbl 0015.35604
[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. MR 1796983
[6] Goebel, K., Kirk, W. A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. 35 (1972), 171–174. MR 0298500 | Zbl 0256.47045
[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. MR 1014125
[8] Ishikawa, S.: Fixed points by new iteration method. Proc. Amer. Math. Soc. 44 (1974), 147–150. MR 0336469
[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. MR 2218570 | 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. MR 2304002 | Zbl 1151.47060
[11] Mann, W. R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4 (1953), 506–510. MR 0054846 | Zbl 0050.11603
[12] Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967), 591–597. MR 0211301 | Zbl 0179.19902
[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. MR 1791754 | Zbl 0971.47038
[14] Schu, J.: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158, 2 (1991), 407–413. MR 1117571 | Zbl 0734.47036
[15] Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. 43, 1 (1991), 153–159. MR 1086729 | Zbl 0709.47051
[16] Senter, H. F., Dotson, W. G., Jr: Approximating fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. 44 (1974), 375–380. MR 0346608 | Zbl 0299.47032
[17] Tan, K. K., Xu, H. K.: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. 122 (1994), 733–739. MR 1203993 | Zbl 0820.47071
[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. MR 2335967 | Zbl 1155.47315
[19] Xu, B., Noor, M. A.: Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 267 (2002), 444–453. MR 1888015 | Zbl 1011.47039
[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. MR 1938840
Partner of
EuDML logo