## Approximating common fixed points for a finite family of asymptotically nonexpansive mappings using iteration process with errors terms.(English)Zbl 1470.47063

Summary: Let $$X$$ be a real Banach space and $$K$$ a nonempty closed convex subset of $$X$$. Let $$T_i : K \rightarrow K$$ ($$i = 1, 2, \dots, m$$) be $$m$$ asymptotically nonexpansive mappings with sequence $$\{k_n \} \subset [1, \infty)$$, $$\sum_{n = 1}^{\infty}(k_n - 1) < \infty$$, and $$\mathcal{F} = \bigcap_{i = 1}^m F(T_i) \ne \varnothing$$, where $$F$$ is the set of fixed points of $$T_i$$. Suppose that $$\{a_{i n} \}_{n = 1}^{\infty}$$, $$\{b_{i n} \}_{n = 1}^{\infty}$$, $$i = 1,2, \dots, m$$ are appropriate sequences in $$[0,1]$$ and $$\{u_{i n} \}_{n = 1}^{\infty}$$, $$i = 1,2, \dots, m$$ are bounded sequences in $$K$$ such that $$\sum_{n = 1}^{\infty} b_{i n} < \infty$$ for $$i = 1,2, \dots, m$$. We give $$\{x_n \}$$ defined by $$x_1 \in K$$, $$x_{n + 1} = (1 - a_{1 n} - b_{1 n}) y_{n + m - 2} + a_{1 n} T_1^n y_{n + m - 2} + b_{1 n} u_{1 n}, y_{n + m - 2} = (1 - a_{2 n} - b_{2 n}) y_{n + m - 3} + a_{2 n} T_2^n y_{n + m - 3} +b_{2 n} u_{2 n}, \dots, y_{n + 2} = (1 - a_{(m - 2) n} - b_{(m - 2) n}) y_{n + 1} + a_{(m - 2) n} T_{m - 2}^n y_{n + 1} + b_{(m - 2) n} u_{(m - 2) n}, y_{n + 1} = (1 - a_{(m - 1) n} -b_{(m - 1) n}) y_n + a_{(m - 1) n} T_{m - 1}^n y_n + b_{(m - 1) n} u_{(m - 1) n},y_n = (1 - a_{m n} - b_{m n}) x_n + a_{m n} T_m^n x_n + b_{m n} u_{m n}$$, $$m \geq 2$$, $$n \geq 1$$. The purpose of this paper is to study the above iteration scheme for approximating common fixed points of a finite family of asymptotically nonexpansive mappings and to prove weak and some strong convergence theorems for such mappings in real Banach spaces. The results obtained in this paper extend and improve some results in the existing literature.

### MSC:

 47J26 Fixed-point iterations 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text:

### References:

  Goebel, K.; Kirk, W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proceedings of the American Mathmatical Society, 35, 171-174 (1972) · Zbl 0256.47045  Chang, S. C.; Cho, Y. J.; Zhou, H., Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings, Journal of the Korean Mathematical Society, 38, 6, 1245-1260 (2001) · Zbl 1020.47059  Cho, Y. J.; Zhou, H.; Guo, G., Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Computers and Mathematics with Applications, 47, 4-5, 707-717 (2004) · Zbl 1081.47063  Noor, M. A., New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications, 251, 1, 217-229 (2000) · Zbl 0964.49007  Osilike, M. O.; Aniagbosor, S. C., Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Mathematical and Computer Modelling, 32, 10, 1181-1191 (2000) · Zbl 0971.47038  Plubtieng, S.; Wangkeeree, R.; Punpaeng, R., On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings, Journal of Mathematical Analysis and Applications, 322, 2, 1018-1029 (2006) · Zbl 1097.47057  Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bulletin of the Australian Mathematical Society, 43, 1, 153-159 (1991) · Zbl 0709.47051  Xu, B.; Noor, M. A., Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 267, 2, 444-453 (2002) · Zbl 1011.47039  Chidume, C. E.; Ali, B., Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 326, 2, 960-973 (2007) · Zbl 1112.47053  Yıldırım, I.; Özdemir, M., Approximating common fixed points of asymptotically quasi-nonexpansive mappings by a new iterative process, Arabian Journal for Science and Engineering, 36, 3, 393-403 (2011) · Zbl 1308.47082  Quan, J.; Chang, S.-S.; Long, X. J., Approximation common fixed point of asymptotically quasi-nonexpansive-type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications, 2006 (2006) · Zbl 1129.47057  Peng, J. W., On the convergence of finite steps iterative sequences with errors asymptotically nonexpansive mappings, IAENG International Journal of Applied Mathematics, 37, 2 (2007) · Zbl 1229.65096  Kızıltunç, H.; Temir, S., Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces, Computers and Mathematics with Applications, 61, 9, 2480-2489 (2011) · Zbl 1368.47066  Tan, K. K.; Xu, H. K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, Journal of Mathematical Analysis and Applications, 178, 2, 301-308 (1993) · Zbl 0895.47048  Xu, H.-K., Inequalities in Banach spaces with applications, Nonlinear Analysis: Theory, Methods & Applications, 16, 12, 1127-1138 (1991) · Zbl 0757.46033
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.