Explicit averaging cyclic algorithm for common fixed points of a finite family of asymptotically strictly pseudocontract
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Explicit averaging cyclic algorithm for common fixed points of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces Ying Zhang1,2* and Zhiwei Xie3 *
Correspondence: [email protected] 1 School of Mathematics and Physics, North China Electric Power University, Baoding, Hebei 071003, P.R. China 2 School of Economics, Renmin University of China, Beijing, 100872, P.R. China Full list of author information is available at the end of the article
Abstract Let E be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty, closed and convex subset of E. We obtain a weak convergence theorem of the explicit averaging cyclic algorithm for a finite family of asymptotically strictly pseudocontractive mappings of K under suitable control conditions, and elicit a necessary and sufficient condition that guarantees strong convergence of an explicit averaging cyclic process to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces. The results of this paper are interesting extensions of those known results. MSC: 47H09; 47H10 Keywords: asymptotically strictly pseudocontractive mappings; weak and strong convergence; explicit averaging cyclic algorithm; fixed points; q-uniformly smooth Banach spaces
1 Introduction Let E and E* be a real Banach space and the dual space of E, respectively. Let Jq (q > ) * denote the generalized duality mapping from E into E given by Jq (x) = {f ∈ E* : x, f = xq and f = xq– } for all x ∈ E, where ·, · denotes the generalized duality pairing between E and E* . In particular, J is called the normalized duality mapping and it is usually denoted by J. If E is smooth or E* is strictly convex, then Jq is single-valued. In the sequel, we will denote the single-valued generalized duality mapping by jq . Let K be a nonempty subset of E. A mapping T : K → K is called asymptotically κ-strictly pseudocontractive with sequence {κn }∞ n= ⊆ [, ∞) such that limn→∞ κn = (see, e.g., [–]) if for all x, y ∈ K , there exist a constant κ ∈ [, ) and jq (x – y) ∈ Jq (x – y) such that n q T x – T n y, jq (x – y) ≤ κn x – yq – κ x – y – T n x – T n y ,
∀n ≥ .
()
© 2012 Zhang and Xie; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zhang and Xie Fixed Point Theory and Applications 2012, 2012:167 http://www.fixedpointtheoryandapplications.com/content/2012/1/167
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If I denotes the identity operator, then () can be written in the form
I – T n x – I – T n y, jq (x – y) q ≥ κ I – T n x – I – T n y – (κn – )x – yq .
()
The class of asymptotically κ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou []. I
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