New iterative methods for a common solution of fixed points for pseudo-contractive mappings and variational inequalities
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RESEARCH
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New iterative methods for a common solution of fixed points for pseudo-contractive mappings and variational inequalities Rabian Wangkeeree1 and Kamonrat Nammanee2* *
Correspondence: [email protected] 2 Department of Mathematics, School of Science, University of Phayao, Phayao, 56000, Thailand Full list of author information is available at the end of the article
Abstract In this paper, we introduce three iterative methods for finding a common element of the set of fixed points for a continuous pseudo-contractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings. Strong convergence theorems for the proposed iterative methods are proved. Our results improve and extend some recent results in the literature. MSC: 47H05; 47H09; 47J25; 65J15 Keywords: pseudo-contractive mapping; monotone mapping; strong convergence theorem
1 Introduction The theory of variational inequalities represents, in fact, a very natural generalization of the theory of boundary value problems and allows us to consider new problems arising from many fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. While the variational theory of boundary value problems has its starting point in the method of orthogonal projection, the theory of variational inequalities has its starting point in the projection on a convex set. Let C be a nonempty closed and convex subset of a real Hilbert space H. The classical variational inequality problem is to find u ∈ C such that v – u, Au ≥ for all v ∈ C, where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by VI(C, A). The variational inequality problem has been extensively studied in the literature; see [–] and the references therein. In the context of the variational inequality problem, this implies that u ∈ VI(C, A) ⇔ u = PC (u – λAu), ∀λ > , where PC is a metric projection of H into C. Let A be a mapping from C to H, then A is called monotone if and only if for each x, y ∈ C, x – y, Ax – Ax ≥ .
(.)
An operator A is said to be strongly positive on H if there exists a constant γ > such that Ax, x ≥ γ x ,
∀x ∈ H.
© 2013 Wangkeeree and Nammanee; 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.
Wangkeeree and Nammanee Fixed Point Theory and Applications 2013, 2013:233 http://www.fixedpointtheoryandapplications.com/content/2013/1/233
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A mapping A of C into itself is called L-Lipschitz continuous if there exits a positive number L such that Ax – Ay ≤ Lx – y,
∀x, y ∈ C.
A mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that x – y, Ax – Ay ≥ αAx – Ay for all x, y ∈ C; see [–]. If A is
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