A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces
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Research Article A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces Urailuk Singthong1 and Suthep Suantai1, 2 1 2
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
Correspondence should be addressed to Suthep Suantai, [email protected] Received 10 February 2010; Revised 21 June 2010; Accepted 15 July 2010 Academic Editor: Massimo Furi Copyright q 2010 U. Singthong and S. Suantai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a new general iterative method by using the K-mapping for finding a common fixed point of a finite family of nonexpansive mappings in the framework of Hilbert spaces. A strong convergence theorem of the purposed iterative method is established under some certain control conditions. Our results improve and extend the results announced by many others.
1. Introduction Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. A mapping T of C into itself is called nonexpansive if T x − T y ≤ x − y for all x, y ∈ C. A point x ∈ C is called a fixed point of T provided that T x x. We denote by FT the set of fixed points of T i.e., FT {x ∈ H : T x x}. Recall that a self-mapping f : C → C is a contraction on C, if there exists a constant α ∈ 0, 1 such that fx − fy ≤ αx − y for all x, y ∈ C. A bounded linear operator A on H is called strongly positive with coefficient γ if there is a constant γ > 0 with the property Ax, x ≥ γx2 ,
∀x ∈ H.
1.1
In 1953, Mann 1 introduced a well-known classical iteration to approximate a fixed point of a nonexpansive mapping. This iteration is defined as xn1 αn xn 1 − αn T xn ,
n ≥ 0,
1.2
2
Fixed Point Theory and Applications
where the initial guess x0 is taken in C arbitrarily, and the sequence {αn }∞ n0 is in the interval 0, 1. But Mann’s iteration process has only weak convergence, even in a Hilbert space setting. In general for example, Reich 2 showed that if E is a uniformly convex Banach space and has a Frehet differentiable norm and if the sequence {αn } is such that Σ∞ n1 αn 1 − αn ∞, then the sequence {xn } generated by process 1.2 converges weakly to a point in FT . Therefore, many authors try to modify Mann’s iteration process to have strong convergence. In 2005, Kim and Xu 3 introduced the following iteration process: x0 x ∈ C arbitrarily chosen, yn βn xn 1 − βn T xn ,
1.3
xn1 αn u 1 − αn yn . They proved in a uniformly smooth Banach space that the sequence {xn } defined by 1.3 converges strongly to a fixed point of T under some appropriate conditions on {αn } and {βn }. In 2008, Yao et al. 4 alsomodified Mann’s iterative scheme 1.2 to get a strong convergence
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