An Extragradient Method for Finding Minimum-Norm Solution of the Split Equilibrium Problem
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An Extragradient Method for Finding Minimum-Norm Solution of the Split Equilibrium Problem Tran Viet Anh1
Received: 6 January 2016 / Revised: 13 June 2016 / Accepted: 14 June 2016 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016
Abstract The purpose of this paper is to find minimum-norm solutions of the split equilibrium problem. This problem is motivated by the least-squares solution to the constrained linear inverse problem. By using the extragradient method, we derive the strong convergence of an iterative algorithm to the minimum-norm solution of the split equilibrium problem. As special cases, minimum-norm solutions of the split variational inequality problem and the split feasibility problem can be found. Keywords Split equilibrium problem · Minimum-norm solution · Strong convergence Mathematics Subject Classification (2010) 49M37 · 90C26 · 65K15
1 Introduction Let C and Q be two nonempty closed convex subsets of two real Hilbert spaces H1 and H2 , respectively, and let A : H1 −→ H2 be a bounded linear operator. Let f : C × C −→ R and g : Q × Q −→ R be two functions, then the split equilibrium problem (in short, SEP) can be formulated as Find x ∗ ∈ C : f (x ∗ , x) ≥ 0 ∀x ∈ C (1) such that (2) y ∗ = Ax ∗ ∈ Q : g(y ∗ , y) ≥ 0 ∀y ∈ Q, which has been first introduced and studied by Zhenhua He [15]. We will denote by Sol(C, f ) and Sol(Q, g) the solution sets of (1) and (2), respectively; then, the SEP becomes the problem of finding x ∗ ∈ Sol(C, f ) such that Ax ∗ ∈ Sol(Q, g). If we consider only the problem (1), then (1) is a classical equilibrium problem. If H1 = H2 , C = Q and A is the identity mapping in H1 , then the SEP becomes the problem of finding a common solution
Tran Viet Anh
[email protected] 1
Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam
T. V. Anh
of two equilibrium problems (1) and (2), which was studied by many authors, for example [1, 2]. From (1) and (2), we can see that the SEP is the problem to find a solution x ∗ of the equilibrium problem (1) in H1 so that the image y ∗ = Ax ∗ , under a given bounded linear operator A, is a solution of another equilibrium problem (2) in another space H2 . It is well known (see, for instance, [4–7, 17, 19]) that the equilibrium problem (1) includes, as special cases, some classes of the optimization problem, Kakutani fixed points, the variational inequality, the Nash equilibrium problem in noncooperative games theory, and minimax problems. So the equilibrium problem (1) is very important in the field of applied mathematics. However, in general, some equilibrium problems belong to different spaces, so the SEP is quite general. The SEP could enable us to split the solution between two different subsets of spaces so that the image of a solution of one equilibrium problem, under a given bounded linear operator, is a solution of another equilibrium problem. When f (x, y) = F (x), y − x, g(u, v) = G(u), v −
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