Strong convergence of an iterative process for a family of strictly pseudocontractive mappings
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RESEARCH
Open Access
Strong convergence of an iterative process for a family of strictly pseudocontractive mappings Yuan Qing1 , Sun Young Cho2 and Meijuan Shang3,4* *
Correspondence: [email protected] 3 Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing, 100044, China 4 Department of Mathematics, Shijiazhuang University, Shijiazhuang, 050035, China Full list of author information is available at the end of the article
Abstract In this article, fixed point problems of a family of strictly pseudocontractive mappings are investigated based on an iterative process. Strong convergence of the iterative process is obtained in a real 2-uniformly Banach space. MSC: 47H09; 47J05; 47J25 Keywords: accretive operator; iterative process; fixed point; nonexpansive mapping; zero point
1 Introduction and preliminaries Throughout this paper, we always assume that E is a real Banach space. Let E∗ be the dual ∗ space of E. Let Jq (q > ) denote the generalized duality mapping from E into E given by Jq (x) = f ∗ ∈ E∗ : x, f ∗ = xq , f ∗ = xq– ,
∀x ∈ E,
where ·, · denotes the generalized duality pairing. In particular, J is called the normalized duality mapping, which is usually denoted by J. In this paper, we use j to denote the singlevalued normalized duality mapping. It is known that Jq (x) = xq– J(x) if x = . If E is a Hilbert space, then J = I, the identity mapping. Further, we have the following properties of the generalized duality mapping Jq : () Jq (tx) = t q– Jq (x) for all x ∈ E and t ∈ [, ∞); () Jq (–x) = –Jq (x) for all x ∈ E. A Banach space E is said to be smooth if the limit lim
t→
x + ty – x t
exists for all x, y ∈ UE . It is also said to be uniformly smooth if the limit is attained uniformly for all x, y ∈ UE . The norm of E is said to be Fréchet differentiable if, for any x ∈ UE , the above limit is attained uniformly for all y ∈ UE . The modulus of smoothness of E is the function ρE : [, ∞) → [, ∞) defined by ρE (τ ) = sup
x + y + x – y – : x ≤ , y ≤ τ ,
∀τ ≥ .
© 2013 Qing et al.; 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.
Qing et al. Fixed Point Theory and Applications 2013, 2013:117 http://www.fixedpointtheoryandapplications.com/content/2013/1/117
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The Banach space E is uniformly smooth if and only if limτ →∞ ρEτ(τ ) = . Let q > . The Banach space E is said to be q-uniformly smooth if there exists a constant c > such that ρE (τ ) ≤ cτ q . It is shown in [] that there is no Banach space which is q-uniformly smooth p with q > . Hilbert spaces, Lp (or lp ) spaces and Sobolev space Wm , where p ≥ , are uniformly smooth. Let C be a nonempty closed convex subset of E and let T : C → C be a mapping. In this paper, we use F(T) to denote the fixed poin
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