The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert
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The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert space Jun Yang1,2 · Hongwei Liu1 Received: 8 October 2018 / Accepted: 28 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, we first introduce and analyze a new algorithm for solving equilibrium problems involving Lipschitz-type and pseudomonotone bifunctions in real Hilbert space. The algorithm uses a new step size, we prove the iterative sequence generated by the algorithm converge strongly to a common solution of equilibrium problem and a fixed point problem without the knowledge of the Lipschitz-type constants of bifunction. Finally, another similar algorithm is proposed and numerical experiments are reported to illustrate the efficiency of the proposed algorithms. Keywords Equilibrium problems · Pseudomonotone bifunction · Subgradient extragradient method · Convex set
1 Introduction In this paper, we consider the equilibrium problems (E P) of find x ∗ ∈ C such that f (x ∗ , y) ≥ 0, ∀y ∈ C,
(1)
where C is a nonempty closed convex subset in a real Hilbert space H , f : H × H −→ R is a bifunction. The set of solutions of (1) is denoted by E P( f ). This problem is also known as the Ky Fan’s inequality due to his contribution to this field [1]. It unifies many important mathematical problems, such as optimization problems, com-
B
Jun Yang [email protected] Hongwei Liu [email protected]
1
School of Mathematics and Statistics, Xidian University, Xi’an 710126, Shaanxi, China
2
School of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000, Shaanxi, China
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J. Yang, H. Liu
plementary problems, variational inequality problems, Nash equilibrium problems, fixed point problems [2–4]. In recent decades, many methods have been proposed and analyzed for approximating solution of equilibrium problems [5–13]. One of the most common methods is proximal point method [5,6], but the method cannot be adapted to peudomonotone equilibrium problems [7]. Another fundamental method for equilibrium problem is the extragradient-like methods [6,7,9–12]. Some known methods use step sizes which depend on the Lipschitz-type constants of the bifunctions [6,12]. That fact can make some restrictions in applications because the Lipschitz-type constants are often unknown or difficult to estimate. Recently, iterative methods for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of operators in a real Hilbert space have further developed by many authors [14–16]. Very recently, Dang [8,10,11] proposed algorithms which use a step size sequence for solving strong pseudomonotone equilibrium problems in real Hilbert space. The main purpose of this paper is to propose a new step size for finding a common element of the set of fixed points of a quasinonexpansive mapping and the set of solutions of equilibrium problems involving pseudomonotone and Lipschitz-type bifunctions. The paper is org
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