## Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces.(English)Zbl 1141.47039

The authors prove the following theorems in real uniformly convex Banach space $$E$$. (1) A weak convergence theorem for finite families of asymptotically nonexpansive mappings in the case where the dual space $$E^*$$ of $$E$$ satisfies the Kadec-Klee property. (2) A strong convergence theorem of one member of the family of asymptotically nonexpansive maps $$\{T_i\}$$ satisfies a condition weak than semicompactness.
These theorems generalize and improve results of S. H. Khan and H. Fukhar-ud-din [Nonlinear Anal., Theory Methods Appl. 61, No. 8 (A), 1295–1301 (2005; Zbl 1086.47050)], N. Shahzad [Nonlinear Anal., Theory Methods Appl. 61, No. 6 (A), 1031–1039 (2005; Zbl 1089.47058)] and N. Shahzad and R. Al-Dubiban [Georgian Math. J. 13, No. 3, 529–537 (2006; Zbl 1136.47049)].

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H05 Monotone operators and generalizations

### Citations:

Zbl 1086.47050; Zbl 1089.47058; Zbl 1136.47049
Full Text:

### References:

 [1] Bauschke, H.H., The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. math. anal. appl., 202, 150-159, (1996) · Zbl 0956.47024 [2] Chidume, C.E., Nonexpansive mappings, generalisations and iterative algorithms, (), 383-421 · Zbl 1057.47003 [3] Chidume, C.E.; Ofoedu, E.U.; Zegeye, H., Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. math. anal. appl., 280, 364-374, (2003) · Zbl 1057.47071 [4] Chidume, C.E., Bashir ali, approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, J. math. anal. appl., 326, 2, 960-973, (2007) · Zbl 1112.47053 [5] Chidume, C.E.; Zegeye, H.; Shahzad, N., Convergence theorems for a common fixed point of finite family of nonself nonexpansive mappings, Fixed point theory appl., 1-9, (2005) · Zbl 1106.47054 [6] Cioranescu, I., Geometry of Banach spaces, duality mappings and nonlinear problems, (1990), Kluwer Academic Dordrecht · Zbl 0712.47043 [7] Das, G.; Debata, J.P., Fixed points of quasi-nonexpansive mappings, Indian J. pure appl. math., 17, 1263-1269, (1986) · Zbl 0605.47054 [8] Diestel, J., Geometry of Banach spaces—selected topics, Lecture notes in math., vol. 485, (1975), Springer New York · Zbl 0307.46009 [9] Falset, J.G.; Kaczor, W.; Kuzumow, T.; Reich, S., Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear anal., 43, 377-401, (2001) · Zbl 0983.47040 [10] Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. amer. math. soc., 35, 171-174, (1972) · Zbl 0256.47045 [11] Ishikawa, S., Fixed point by new iteration method, Proc. amer. math. soc., 44, 147-150, (1974) · Zbl 0286.47036 [12] Jung, J.S., Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. math. anal. appl., 302, 509-520, (2005) · Zbl 1062.47069 [13] Jung, J.S.; Cho, Y.J.; Agarwal, R.P., Iterative schemes with some control conditions for family of finite nonexpansive mappings in Banach spaces, Fixed point theory appl., 2, 125-135, (2005) · Zbl 1109.47056 [14] Kaczor, W., Weak convergence of almost orbits of asymptotically nonexpansive semigroups, J. math. anal. appl., 272, 565-574, (2002) · Zbl 1058.47049 [15] Khan, S.H.; Fukharu-ud-din, H., Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear anal., 61, 1295-1301, (2005) · Zbl 1086.47050 [16] Mann, W.R., Mean value method in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603 [17] Oka, H., A nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach space, Proc. Japan acad. ser. A, 65, 284-287, (1998) · Zbl 0762.47027 [18] O’Hara, J.G.; Pillay, P.; Xu, H.K., Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear anal., 54, 1417-1426, (2003) · Zbl 1052.47049 [19] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026 [20] Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. math. anal. appl., 158, 407-413, (1991) · Zbl 0734.47036 [21] Schu, J., Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. austral. math. soc., 43, 153-159, (1991) · Zbl 0709.47051 [22] Senter, H.F.; Dotson, W.G., Approximating fixed points of nonexpansive mappings, Proc. amer. math. soc., 44, 2, 375-380, (1974) · Zbl 0299.47032 [23] Shahzad, N., Approximating fixed points of nonself nonexpansive mappings in Banach spaces, Nonlinear anal., 61, 1031-1039, (2005) · Zbl 1089.47058 [24] N. Shahzad, R. Al-dubiban, Approximating fixed points of nonexpansive mappings in Banach spaces, Georgian Math. J., in press [25] Suzuki, T., Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 227-239, (2005) · Zbl 1068.47085 [26] Tan, K.K.; Xu, H.K., Approximating fixed points of nonexpansive mappings by Ishikawa iteration process, J. math. anal. appl., 178, 301-308, (1993) · Zbl 0895.47048 [27] Tan, K.K.; Xu, H.K., Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. amer. math. soc., 122, 733-739, (1994) · Zbl 0820.47071 [28] Zhou, H.; Wei, L.; Cho, Y.J., Strong convergence theorems on an iterative method for family of finite nonexpansive mappings in reflexive Banach spaces, Appl. math. comput., 173, 196-212, (2006) · Zbl 1100.65049
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.