A New Simple Parallel Iteration Method for a Class of Variational Inequalities
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A New Simple Parallel Iteration Method for a Class of Variational Inequalities Nguyen Song Ha1 · Nguyen Buong2 · Nguyen Thi Thu Thuy1
Received: 6 November 2016 / Revised: 7 June 2017 / Accepted: 8 June 2017 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017
Abstract In this paper, we propose a new simple parallel iterative method to find a solution for variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gˆateaux differentiable norm. Our parallel iterative method is simpler than the one proposed by Buong et al. (Numer. Algorithms 72, 467–481 2016). An iterative method of Halpern type for common zeros of an infinite family of m-accretive mappings is shown as a special case of our result. Two numerical examples are also given to illustrate the effectiveness and superiority of the proposed algorithm. Keywords Fixed point · Nonexpansive and accretive mapping · Variational inequality · Common zero of accretive mappings · Iterative algorithms Mathematics Subject Classification (2010) 47J05 · 47H09 · 49J30
Nguyen Song Ha
[email protected] Nguyen Buong [email protected] Nguyen Thi Thu Thuy [email protected] 1
Thainguyen College of Sciences, Thainguyen University, Thai Nguyen, Vietnam
2
Institute of Information Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
N. S. Ha et al.
1 Introduction Let E be a Banach space with the dual space E ∗ . For the sake of simplicity, the norms of E and E ∗ are denoted by the symbol .. We use x, x ∗ instead of x ∗ (x) for x ∗ ∈ E ∗ and ∗ x ∈ E. A mapping J from E into 2E defined by J (x) = {x ∗ ∈ E ∗ : x, x ∗ = xx ∗
and x ∗ = x},
x∈E
is called a normalized duality mapping of E. Let F : E → E be a single-valued mapping. Then, F : E → E is said to be (i)
(ii)
accretive (see [9, p. 115]) if for each x, y ∈ E, there exists j (x − y) ∈ J (x − y) such that F (x) − F (y), j (x − y) ≥ 0; η-strongly accretive (see [9, p. 115]) if there exists some η > 0 such that for each x, y ∈ E, there exists j (x − y) ∈ J (x − y) such that F (x) − F (y), j (x − y) ≥ ηx − y2 ;
(iii)
m-accretive (see [9, p. 115]) if it is accretive and R(rF +I ) = E for all r > 0, where R(F ) denotes the range of F and I is the identity mapping in E.
If E is a Hilbert space, accretive and η-strongly accretive mappings are also called monotone and η-strongly monotone, respectively. Furthermore, F is called γ -strictly pseudocontractive in the sense of Browder and Petryshyn (see [9, p. 145]) if there exists j (x − y) ∈ J (x − y) such that F (x) − F (y), j (x − y) ≤ x − y2 − γ (I − F )(x) − (I − F )(y)2 holds for x, y ∈ E and for some γ > 0. Clearly, if F is γ -strictly pseudocontractive, F is L-Lipschitz continuous with L = 1 + 1/γ , i.e., F (x) − F (y) ≤ Lx − y for all x, y ∈ E. We denote the set of fixed points of a mapping T : E → E
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