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On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings. (English) Zbl 1086.47044
The authors study the weak and strong convergence of the implicit iteration sequences $\{ x_{n}\} $ defined by: $$\align x_{n}&= \alpha _{n}x_{n-1} + (1-\alpha _{n})T^{k(n)}_{i(n)}x_{n} + u_{n}, \quad n \geq 1\quad\text{and}\\ x_{n}&=\alpha _{n}x_{n-1} + (1- \alpha _{n})T^{k(n)}_{i(n)}x_{n},\quad n \geq 1\endalign$$ to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in real uniformly convex Banach spaces. Their results extend and improve results of several other authors.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
[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] Chang, S. S.; Cho, Y. J.; Zhou, H. Y.: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. J. korean math. Soc. 38, 1245-1260 (2001) · Zbl 1020.47059
[3] Goebel, K.; Kirk, W. A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. amer. Math. soc. 35, 171-174 (1972) · Zbl 0256.47045
[4] Gornicki, J.: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. math. Univ. carolin. 301, 249-252 (1989) · Zbl 0686.47045
[5] Halpern, B.: Fixed points of nonexpansive maps. Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101
[6] Lions, P. L.: Approximation de points fixes de contractions. C. R. Acad. sci. Paris sér. A 284, 1357-1359 (1977) · Zbl 0349.47046
[7] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047
[8] Schu, J.: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. austral. Math. soc. 43, 153-159 (1991) · Zbl 0709.47051
[9] Sun, Z. H.: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. J. math. Anal. appl. 286, 351-358 (2003) · Zbl 1095.47046
[10] Tan, K. K.; Xu, H. K.: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proc. amer. Math. soc. 114, 399-404 (1992) · Zbl 0781.47045
[11] Tan, K. K.; Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process. J. math. Anal. appl. 178, 301-308 (1993) · Zbl 0895.47048
[12] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. math. 58, 486-491 (1992) · Zbl 0797.47036
[13] Xu, H. K.; Ori, M. G.: An implicit iterative process for nonexpansive mappings. Numer. funct. Anal. optim. 22, 767-773 (2001) · Zbl 0999.47043
[14] 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. optim. 23, 911-921 (2002) · Zbl 1041.47048