Abstract
In this paper, we establish some strong convergence theorems of the modified Ishikawa and Mann iterations with errors of uniformly L-Lipschitzian asymptotically pseudocontractive mappings in real Banach spaces. Our results not only provide the new proof method, but also extend the known corresponding results given in (Chang in Proc. Am. Math. Soc. 129:845-853, 2001; Chang et al. in Appl. Math. Lett. 22:121-125, 2009; Goebel and Kirk in Proc. Am. Math. Soc. 35:171-174, 1972; Ofoedu in J. Math. Anal. Appl. 321:722-728, 2006; Schu in J. Math. Anal. Appl. 158:407-413, 1991). In order to get some applications of our results, we also provide specific examples.
Highlights
Introduction and preliminaries LetE be a real Banach space and let J denote the normalized duality mapping from E into E* defined byJ(x) = f ∈ E* : x, f = x = f for all x ∈ E, where E* denotes the dual space of E and ·, · denotes the generalized duality pairing, respectively
A mapping T : D → D is said to be asymptotically nonexpansive with a sequence {kn} ⊂ [, +∞) and limn→∞ kn = if, for all x, y ∈ D, Tnx – Tny ≤ kn x – y for all n ≥
For any x ∈ K, let {xn} be an iterative sequence defined by xn+ = ( – αn)xn + αnT nxn for all n ≥
Summary
[ ] Let H be a Hilbert space, K be a nonempty bounded closed convex subset of H and T : K → K be a completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractive mapping with a sequence {kn} ⊂ [ , +∞) satisfying the following conditions: (a- ) kn → as n → ∞; (a- )
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