Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Abstract

Let E be a real uniformly convex Banach space whose dual space E * satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let T 1 , T 2 , . . . , T m : K → K be asymptotically nonexpansive mappings of K into E with sequences (respectively) {k in } ∞ n=1 satisfying k in → 1 as n → ∞, i = 1, 2, . . . , m, and ∞ n=1 (k in − 1) < ∞. For arbitrary ∈ (0, 1), let {α in } ∞ n=1 be a sequence in [ , 1 − ], for each i ∈ {1, 2, . . . , m} (respectively). Let {x n } be a sequence generated for m 2 byLet m i=1 F (T i ) = ∅. Then, {x n } converges weakly to a common fixed point of the family {T i } m i=1 . Under some appropriate condition on the family {T i } m i=1 , a strong convergence theorem is also proved.

Random fixed point theorems for a finite family of asymptotically quasi-nonexpansive in the intermediate sense random operators

“…OE0; 1/ with f .0/ D 0, f .r/ > 0 for all r 2 .0; 1/ such that for all x 2 C , Condition (B) reduces to condition (A) when all but one of the mappings T i are identities. Also conditions (B) and (C) are equivalent (see [6]). …”

Approximating fixed points of total asymptotically nonexpansive mappings

“…Iterative techniques are the main tool for approximating fixed points of nonexpansive mappings and asymptotically nonexpansive mappings, and it has been studied by various authors using Krasnosel'skii-Mann and Ishikawa schemes (see, e.g., [12,13,15,20,21,25,[27][28][29][30][31][32][33][34][35][36][37]). …”

Weak and Strong Convergence of Multistep Iteration for Finite Family of Asymptotically Nonexpansive Mappings

“…Then, observing S n+m x n = x n+m , we get (2) Theorem 3.4 in [1] is also a special case of Theorem 2.9 with γ (i) n = 0 for i = 1,2,...,N.…”

“…Very recently, Chidume and Ali [1] considered multistep scheme for finite family of asymptotically nonexpansive mappings and gave weak convergence theorems for this scheme in a uniformly convex Banach space whose the dual space satisfies the Kadec-Klee property. They also proved a strong convergence theorem under some appropriate conditions on finite family of asymptotically nonexpansive mappings.…”