Another weak convergence theorems for accretive mappings in banach spaces
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Another weak convergence theorems for accretive mappings in banach spaces Satit Saejung*, Kanokwan Wongchan and Pakorn Yotkaew * Correspondence: [email protected]. th Department of Mathematics, Faculty of Science, Khon kaen University, Khon kaen 40002, Thailand
Abstract We present two weak convergence theorems for inverse strongly accretive mappings in Banach spaces, which are supplements to the recent result of Aoyama et al. [Fixed Point Theory Appl. (2006), Art. ID 35390, 13pp.]. 2000 MSC: 47H10; 47J25. Keywords: weak convergence theorem, accretive mapping, Banach space
1. Introduction Let E be a real Banach space with the dual space E*. We write 〈x, x* 〉 for the value of a functional x*Î E* at x Î E. The normalized duality mapping is the mapping J : E ® 2E* given by Jx = {x∗ ∈ E∗ : x, x∗ = ||x||2 = ||x∗ ||2 }
(x ∈ E).
In this paper, we assume that E is smooth, that is, limt→0 ||x+tx||−||x|| exists for all x, y t Î E with ||x|| = ||y|| = 1. This implies that J is single-valued and we do consider the singleton Jx as an element in E*. For a closed convex subset C of a (smooth) Banach space E, the variational inequality problem for a mapping A : C ® E is the problem of finding an element u Î C such that Au, J(v − u) ≥ 0
for all v ∈ C.
The set of solutions of the problem above is denoted by S(C, A). It is noted that if C = E, then S(C, A) = A-10 := {x Î E : Ax = 0}. This problem was studied by Stampacchia (see, for example, [1,2]). The applicability of the theory has been expanded to various problems from economics, finance, optimization and game theory. Gol’shteĭn and Tret’yakov [3] proved the following result in the finite dimensional space ℝN. Theorem 1.1. Let a > 0, and let A : ℝN ® ℝN be an a-inverse strongly monotone mapping, that is, 〈Ax - Ay, × - y〉 ≥ a||Ax - Ay||2 for all x, y Î ℝN. Suppose that {xn} is a sequence in ℝN defined iteratively by x1 Î ℝN and xn+1 = xn − λn Axn ,
where {ln}⊂ [a, b] ⊂ (0, 2a). If A-1 0 ≠ ∅, then {xn} converges to some element of A-10. The result above was generalized to the framework of Hilbert spaces by Iiduka et al. [4]. Note that every Hilbert space is uniformly convex and 2-uniformly smooth (the related © 2011 Saejung 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.
Saejung et al. Fixed Point Theory and Applications 2011, 2011:26 http://www.fixedpointtheoryandapplications.com/content/2011/1/26
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definitions will be given in the next section). Aoyama et al. [[5], Theorem 3.1] proved the following result. Theorem 1.2. Let E be a uniformly convex and 2-uniformly smooth Banach space with the uniform smoothness constant K, and let C be a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C, let a > 0 and let A : C ® E be an a-inverse strongly accretive
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