Modified multistep iterative process for some common fixed point of a finite family of nonself asymptotically nonexpansive mappings

Let EE be a real uniformly convex Banach space and KK be a nonempty closed convex subset of EE which is also a nonexpansive retract of EE. Let Ti:K→E(i=1,2,…,r) be nonself asymptotically nonexpansive mappings. It is proved that the modified multistep iterative sequence converges weakly and strongly to some common fixed point of {Ti}i=1r under suitable conditions. The results of this paper improve and extend the corresponding results of [J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991) 407–413; M.O. Osilike, S.C. Aniagbosor, Weak and strong convergence theormes for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32 (2000) 1181–1191; W. Takahashi, G.E. Kim, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces, Nonlinear Anal. 3 (32) (1998) 447–454; C.E. Chidume, E.U. Ofoedu, H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 280 (2003) 364–374; L. Wang, Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl. 323 (2006) 550–557; B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444–453]. On the other hand, it is shown that the necessary and sufficient conditions for the strong convergence of the modified multistep iterative sequence to some common fixed points of {Ti}i=1r.