A new iterative method for a common solution of fixed points for pseudo-contractive mappings and variational inequalitie
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A new iterative method for a common solution of fixed points for pseudo-contractive mappings and variational inequalities Tanom Chamnarnpan and Poom Kumam* * Correspondence: poom. [email protected] Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand
Abstract In this article, we introduce a new iterative scheme 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 for the proposed iterative scheme is proved. Our results improve and extend some recent results in the literature. 2000 Mathematics Subject Classification: 46C05; 47H09; 47H10. Keywords: monotone mapping, nonexpansive mapping, pseudo-contractive mappings, variational inequality
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 a u Î C such that 〈v-u, Au〉 ≥ 0 for all v Î 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 [1-5] and the reference therein. In the context of the variational inequality problem, this implies that u Î VI(C, A) ⇔ u = PC(u - lAu), ∀l > 0, 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 Î C, x − y, Ax − Ay ≥ 0.
(1:1)
An operator A is said to be strongly positive on H if there exists a constant γ¯ > 0 such that
© 2012 Chamnarnpan and Kumam; 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.
Chamnarnpan and Kumam Fixed Point Theory and Applications 2012, 2012:67 http://www.fixedpointtheoryandapplications.com/content/2012/1/67
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Ax, x ≥ γ¯ x2 , ∀x ∈ H.
A mapping A of C into itself is called L-Lipschitz continuous if there exits a positive and number L such that Ax − Ay ≤ L x − y , ∀x, y ∈ C. A mapping A of C into H is called a-inverse-strongly monotone if there exists a positive real number a such that 2 x − y, Ax − Ay ≥ α Ax − Ay
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