On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings

Abstract

In this paper, several weak and strong convergence theorems are established for a modified Noor iterative scheme with errors for three asymptotically nonexpansive mappings in Banach spaces. Mann-type, Ishikawa-type, and Noor-type iterations are covered by the new iteration scheme. Our results extend and improve the recently announced ones [B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444-453; Y.J. Cho, H. Zhou, G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004) 707-717] and many others.

New iterative schemes for a finite family of nonself uniformly quasi-Lipschitzian mappings in Banach spaces

“…Fixed points iterative technique for (self or nonself) asymptotically nonexpansive mappings in Banach spaces, including Mann type iteration, Ishikawa type iteration, and three-step type iteration, have been studied by many authors (see, e.g., [2][3][4][5][6][7][8]). Recently, Khan et al [9] introduced an iterative scheme (which generalizes Mann iteration, Ishikawa iteration, and three-step iteration) for a finite family of asymptotically quasi-nonexpansive self-mappings {T i : i I}: C C, where I = {1, 2, .…”

Fixed point theorems of finite family with errors for asymptotically quasi-nonexpansive mappings in convex metric spaces

“…Moreover, Suantai [18] gave weak and strong convergence theorems for a new three step iterative scheme of asymptotically nonexpansive mappings. More recently, Plubtieng et al [16] introduced three step iterative scheme with errors for three asymptotically nonexpansive mapping and established strong convergence of this scheme to common fixed point of three asymptotically nonexpansive mappings.…”

Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces

“…Fixed-point iterations process for nonexpansive and asymptotically nonexpansive mappings in Banach spaces have been studied extensively by various authors [1][2][3][4][5][6][7][8][9][10][11][12][13]. In 1991, Schu [4] considered the following modified Mann iteration process for an asymptotically nonexpansive map T on C and a sequence {a n } in [0, 1]:…”

“…Since then, Schu's iteration process (1.1) has been widely used to approximate fixed points of asymptotically nonexpansive mappings in Hilbert spaces or Banach spaces [7,8,[10][11][12][13]. Noor, in 2000, introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces [6].…”

“…Noor, in 2000, introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces [6]. Later, Xu and Noor [7], Cho et al [8], Suantai [9], Plubtieng et al [12] studied the convergence of the three-step iterations for asymptotically nonexpansive mappings in a uniformly convex Banach space which satisfies Opial's condition or whose norm is Fréchet differentiable.…”