Abstract
Abstract In this article, we introduce a new three-step iterative scheme for the mappings which are asymptotically nonexpansive in the intermediate sense in Banach spaces. Weak convergence theorem is established for this three-step iterative scheme in a uniformly convex Banach space that satisfies Opial's condition or whose dual space has the Kadec-Klee property. Furthermore, we give an example of the nonself mapping which is asymptotically nonexpansive in the intermediate sense but not asymptotically nonexpansive. The results obtained in this article extend and improve many recent results in this area. AMS classification: 47H10; 47H09; 46B20.
Highlights
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]
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
A nonself mapping T : C ® X is called asymptotically nonexpansive in the intermediate sense if T is continuous and the following inequality holds: lim sup sup ( T(PT)n−1x − T(PT)n−1y − x − y ) ≤ 0
Summary
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 2003, Chidume et al [16] introduced the following modified Mann iteration process and got the convergence theorems for asymptotically nonexpansive nonself-mapping: x1 ∈ C, xn+1 = P[αnxn + (1 − αn)T(PT)n−1xn], n ≥ 1.
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