Abstract
A finite-step iteration sequence for two finite families of asymptotically nonexpansive mappings is introduced and the weak and strong convergence theorems are proved in Banach space. The results presented in the paper generalize and unify some important known results of relevant scholars.
Highlights
Introduction and PreliminariesThroughout this work, we assume that E is a real Banach space and K is a nonempty subset of E
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972 as an important generalization of the class of nonexpansive self-mappings, who proved that if K is a nonempty closed convex subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive self-mapping of K, T has a fixed point
In [2], the authors introduced a multistep procedure defined by (2); under some conditions, they proved that the convergence of Mann-Ishikawa iterations is equivalent to the convergence of the multistep iteration in Banach spaces: x1 ∈ E, xn+1 = (1 − βn1) xn + βn1Tyn1, x1 ∈ K, xn+1 = (1 − βn1) xn + βn1T1nyn+m−2, yn+m−2 = (1 − βn2) xn + βn2T2nyn+m−3, n ≥ 1, (3)
Summary
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China A finite-step iteration sequence for two finite families of asymptotically nonexpansive mappings is introduced and the weak and strong convergence theorems are proved in Banach space. The results presented in the paper generalize and unify some important known results of relevant scholars.
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