Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infi
- PDF / 412,201 Bytes
- 16 Pages / 595.276 x 793.701 pts Page_size
- 18 Downloads / 197 Views
RESEARCH
Open Access
Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions 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 an infinite family of istrict pseudo-contractions and a sequence of positive real numbers. By using this mapping, we consider an iterative method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudocontractions. Strong convergence theorem of the purposed iteration is established in the framework of Hilbert spaces. Keywords: nonexpansive mappings, strongly positive operator, generalized equilibrium problem, strict pseudo-contraction, fixed point
1 Introduction Let C be a closed convex subset of a real Hilbert space H, and let G : C × C ® ℝ be a bifunction. We know that the equilibrium problem for a bifunction G is to find x Î C such that G(x, y) ≥ 0
∀y ∈ C.
(1:1)
The set of solutions of (1.1) is denoted by EP(G). Given a mapping T : C ® H, let G (x, y) = 〈Tx, y - x〉 for all x, y Î. Then, z Î EP(G) if and only if 〈Tz, y - z〉 ≥ 0 for all y Î C, i.e., z is a solution of the variational inequality. Let A : C ® H be a nonlinear mapping. The variational inequality problem is to find a u Î C such that v − u, Au ≥ 0
(1:2)
for all v Î C. The set of solutions of the variational inequality is denoted by V I(C, A). Now, we consider the following generalized equilibrium problem: Find z ∈ C such that G(z, y) + Az, y − z ≥ 0,
∀y ∈ C.
(1:3)
The set of such z Î C is denoted by EP(G, A), i.e., EP(G, A) = {z ∈ C : G(z, y) + Az, y − z ≥ 0,
∀y ∈ C.
© 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, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23
Page 2 of 16
In the case of A ≡ 0, EP(G, A) is denoted by EP(G). In the case of G ≡ 0, EP(G, A) is also denoted by V I(C, A). Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics reduce to find a solution of (1.3) (see, for instance, [1]-[3]). 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 − Ay||2
for all x, y Î C. A mapping T with domain D(T) and range R(T) is called nonexpansive if ||Tx − Ty|| ≤ ||x − y||
(1:4)
for all x, y Î D(T) and T is said to be -str
Data Loading...
