New extragradient methods for solving equilibrium problems in Banach spaces
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00096-5 ORIGINAL PAPER
New extragradient methods for solving equilibrium problems in Banach spaces Dang Van Hieu1 · Le Dung Muu2 · Pham Kim Quy3 · Hoang Ngoc Duong3 Received: 5 June 2020 / Accepted: 5 September 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, three new algorithms are proposed for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a 2-uniformly convex and uniformly smooth Banach space. The algorithms are constructed around the 𝜙-proximal mapping associated with cost bifunction. The first algorithm is designed with the prior knowledge of the Lipschitz-type constant of bifunction. This means that the Lipschitz-type constant is an input parameter of the algorithm while the next two algorithms are modified such that they can work without any information of the Lipschitz-type constant, and then they can be implemented more easily. Some convergence theorems are proved under mild conditions. Our results extend and enrich existing algorithms for solving equilibrium problem in Banach spaces. The numerical behavior of the new algorithms is also illustrated via several experiments. Keywords Equilibrium problem · Extragradient method · Pseudomonotone bifunction · Lipschitz-type condition Mathematics Subject Classification 90C33 · 47J20
Communicated by Ti-Jun Xiao. * Dang Van Hieu [email protected] Le Dung Muu [email protected] Pham Kim Quy [email protected] Hoang Ngoc Duong [email protected] 1
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
TIMAS, Thang Long University, Hanoi, Vietnam
3
Department of Basic Sciences, College of Air Force, Nha Trang, Vietnam
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D. Van Hieu et al.
1 Introduction The paper concerns with some new iterative methods for approximating a solution of an equilibrium problem in a Banach space E. Recall that an equilibrium problem (EP) is stated as follows:
Find x∗ ∈ C such that f (x∗ , y) ≥ 0 for all y ∈ C,
(EP)
where C is a nonempty closed convex subset in E, f ∶ C × C → ℜ is a bifunction with f (x, x) = 0 for all x ∈ C . Let us denote by EP(f, C) the solution set of the problem (EP). Mathematically, the problem (EP) has a simple and convenient structure. This problem is quite general, and it unifies several known models in applied sciences as optimization problems, variational inequalities, hemivariational inequalities, fixed point problems and others [3, 12, 15, 28, 35, 43, 44]. In recent years, this problem has received a lot of attention by many authors in both theory and algorithm. Some methods have been proposed for solving the problem (EP) such as the proximal point method [30, 34], the extragradient method [13, 38], the descent method [4–6, 29], the linesearch extragradient method [38], the projected subgradient method [42] and others [7, 36, 48, 51]. The proximal-like method was early introduced in [13] and its convergence was also studied. Re
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