Iterative algorithms for finding a common solution of system of the set of variational inclusion problems and the set of
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RESEARCH
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Iterative algorithms for finding a common solution of system of the set of variational inclusion problems and the set of fixed point problems Atid Kangtunyakarn Correspondence: [email protected] Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
Abstract In this article, we introduce a new mapping generated by infinite family of nonexpansive mapping and infinite real numbers. By means of the new mapping, we prove a strong convergence theorem for finding a common element of the set of fixed point problems of infinite family of nonexpansive mappings and the set of a finite family of variational inclusion problems in Hilbert space. In the last section, we apply our main result to prove a strong convergence theorem for finding a common element of the set of fixed point problems of infinite family of strictly pseudocontractive mappings and the set of finite family of variational inclusion problems. Keywords: nonexpansive mapping, strict pseudo contraction, strongly positive operator, variational inclusion problem, fixed point
1 Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let A : C ® H be a nonlinear mapping and let F : C × C ® ℝ be a bifunction. A mapping T of H into itself is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y Î H. We denote by F (T) the set of fixed points of T (i.e. F(T) = {x Î H : Tx = x}). Goebel and Kirk [1] showed that F(T) is always closed convex and also nonempty provided T has a bounded trajectory. The problem for finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see, e.g., [2,3]). A bounded linear operator A on H is called strongly positive with coefficient γ¯ if there exists a constant γ¯ > 0 with the property Ax, x ≥ γ¯ x2 .
A mapping A of C into H is called inverse-strongly monotone, see [4], if there exists a positive real number a such that x − y, Ax − Ay ≥ α Ax − Ay2
for all x, y Î C. The variational inequality problem is to find a point u Î C such that v − u, Au ≥ 0
for allv ∈ C.
(1:1)
© 2011 Kangtunyakarn; 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.
Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38
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The set of solutions of (1.1) is denoted by V I(C, A). Many authors have studied methods for finding solution of variational inequality problems (see, e.g., [5-8]). In 2008, Qin et al. [9] introduced the following iterative scheme: yn = PC (I − sn A)xn (1:2) xn+1 = αn γ f (Wn xn ) +
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