On the approximation problem of common fixed points for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces

Abstract

The purpose of this paper is to introduce the concept of non-self asymptotically quasi-nonexpansive-type mappings and to construct a iterative sequence to converge to a common fixed point for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces. The results presented in this paper improve and extend the corresponding results in Alber, Chidume and Zegeye [Ya.I. Alber, C.E. Chidume, H. Zegeye, Approximating of total asymptotically nonexpansive mappings, Fixed Point Theory and Applications (2006) 1–20. Article ID10673], Ghosh and Debnath [M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 207 (1997) 96–103], Liu [Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive type mappings, Journal of Mathematical Analysis and Applications 259 (2001) 1–37; Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications 259 (2001) 18–24; Q.H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space, Journal of Mathematical Analysis and Applications 266 (2002) 468–471], Petryshyn [W.V. Petryshyn, T.E. Williamson Jr., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 43 (1973) 459–497], Quan and Chang [J. Quan, S.S. Chang, X.J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications V (2006) 1–38. Article ID 70830], Shahzad and Udomene [N. Shahzad, A. Udomene, Approximating common fixed point of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2006) 1–10. Article ID 18909] Xu [B.L. Xu, M.A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 267 (2002) 444–453], Zhang [S.S. Zhang, Iterative approximation problem of fixed points for asymptotically nonexpansive mappings in Banach spaces, Acta Mathematicae Applicatae Sinica 24 (2001) 236–241] and Zhou and Chang [Y.Y. Zhou, S.S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numerical Functional Analysis and Optimization 23 (2002) 911–921].