CONVERGENCE THEOREMS FOR A FINITE FAMILY OF NONEXPANSIVE AND TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

Abstract

CONVERGENCE THEOREMS FOR A FINITE FAMILY OF NONEXPANSIVE AND TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

Abstract

In this paper, we define and study strong convergence of finite step iteration sequences with errors to a common fixed point for a pair consisting of a finite family of nonexpansive mappings and a finite family of total asymptotically nonexpansive mappings in a nonempty closed convex subset of uniformly convex Banach spaces. The results of this paper can be viewed as an improvement and extension of the corresponding results of F. Gu and Z. He (Multi-step iterative process with errors for common fixed points of a finite family of nonexpansive mappings, Math. Commun. 11 (1), 47–54, 2006), Z. Liu, R. P. Agarwal, C. Feng, and S. M. Kang ( Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings, Acta Univ. Palack. Olomuc. Fac. Rer Nat. Math. 44, 83–96, 2005), Z. Liu, C. Feng, J. S. Ume and S. M. Kang (Weak and strong convergence for common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings, Taiwanese J. Math. 11, 27–42, 2007), S. Saejung and K. Sitthikul (Weak and strong convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings in Banach spaces, Thai J. Math. Special Issue, 15–26, 2008) and others.

Keywords

• Total asymptotically nonexpansive mappings,Common fixed point,Uniformly convex Banach space,Finite-step iteration scheme with errors with respect to a pair of mappings

References

1. Albert, Ya. I., Chidume, C. E. and Zegeye, H. Approximating fixed points of total asymptot- ically nonexpansive mappings, Fixed Point Theory Appl. Article ID 10673, 2006.
2. Bruck, R. E., Kuczumow,T. and Reich, S. Convergence of iterates of asymptotically nonex- pansive mappings in Banach spaces with the uniform Opial property, in: Colloq. Math. 2, 169–179, 1993.
3. Goebel, K. and Kirk, W. A. A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35, 171–174, 1972.
4. Gu, F. and He, Z. Multi-step iterative process with errors for common fixed points of a finite family of nonexpansive mappings, Math. Commun. 11 (1), 47–54, 2006.
5. Kirk, W. A. Fixed point theorems for non-Lipschitzian mappings of asymptotically nonex- pansive type, Israel J. Math. 17, 339–346, 1974.
6. Liu, Z., Agarwal, R. P., Feng, C. and Kang, S. M. Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings, Acta Univ. Palack. Olomuc. Fac. Rer Nat. Math. 44, 83–96, 2005.
7. Liu, Z., Feng, C., Ume, J. S. and Kang, S. M. Weak and strong convergence for common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings, Taiwanese J. Math. 11, 27–42, 2007.
8. Osilike, M. O. and Aniagbosar, S. C. Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32, 1181–1191, 2000 . [9] Saejung, S. and Sitthikul, K. Convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings, Acta Univ. Palack. Olomuc. Fac. Rer Nat. Math. 48, 139–152, 2009.

9. Saejung, S. and Sitthikul, K. Weak and strong convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings in Banach spaces, Thai J. Math. Special Issue, 15–26, 2008.
10. Saluja, G. S. Strong convergence for common fixed points of a pair of quasi-nonexpansive and asymptotically quasi-nonexpansive mappings, Functional Analysis, Approximation and Computation 2 (1), 33–51, 2010.
11. Schu, J. Weak and strong convergence of a fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc. 43, 153–159, 1991.
12. Schu, J. Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158, 407–413, 1991 .
13. Tan, K. K. and Xu, H. K. Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178, 301–308, 1993.
14. Tan, K. K. and Xu, H. K. Fixed point iteration process for asymptotically nonexpansive mappings in Banach Spaces, Proc. Amer. Math. Soc. 122, 733–739, 1994 .
15. Xu, H. K. Existence and convergence for fixed points of mappings of asymptotically nonex- pansive type, Nonlinear Anal. 16, 1139–1146, 1991.