A New Method for Solving Variational Inequalities and Fixed Points Problems of Demi-Contractive Mappings in Hilbert Spac
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A New Method for Solving Variational Inequalities and Fixed Points Problems of Demi-Contractive Mappings in Hilbert Spaces Xue Chen1,2 · Zhong-bao Wang1,2,3 · Zhang-you Chen1,2 Received: 14 October 2019 / Revised: 27 September 2020 / Accepted: 30 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, an efficient algorithm for solving variational inequalities and fixed points problems of demi-contractive mappings is proposed in Hilbert spaces. The algorithm uses variable stepsizes which are updated at each iteration by a cheap computation without any linesearch procedure. Under the assumption that the mapping is pseudomonotone and without prior knowledge of the Lipschitz constant of the underlying operator, the sequence generated by the algorithm is strongly convergent to a common element of the set of fixed points of a demi-contractive mapping and the solution set of variational inequalities. Some experiments are performed to show the numerical behavior of the proposed algorithm and also to compare its performance with those of others. Keywords Variational inequality · Fixed points problems · Demi-contractive mappings · Strong convergence · Pseudomonotonicity Mathematics Subject Classification 65Y05 · 65K15 · 47H05 · 47H10
This work was supported by the National Natural Science Foundation of China (11701479, 11526170,11701478,11771067) and the Chinese Postdoctoral Science Foundation (2018M643434).
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Zhong-bao Wang [email protected] Zhang-you Chen [email protected]
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School of Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan, People’s Republic of China
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National Engineering Laboratory of Integrated Transportation Big Data Application Technology, Chengdu 611756, People’s Republic of China
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School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, People’s Republic of China 0123456789().: V,-vol
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Page 2 of 21
Journal of Scientific Computing
(2020) 85:18
1 Introduction Let H be a real Hilbert space with inner product ·, · and its induced norm · , C be a nonempty, closed and convex subset of H , and F : H → H be a single-valued mapping. Let the notations and → denote weak and strong convergence of sequences respectively. The variational inequalities (VI) are to find x ∗ ∈ C such that F(x ∗ ), y − x ∗ ≥ 0, ∀ y ∈ C.
(1.1)
For simplicity, V I (F, C) denotes the solution set of problem (1.1), i.e., V I (F, C) = {x ∗ ∈ C : F(x ∗ ), y − x ∗ ≥ 0, ∀ y ∈ C}. The study of VI(1.1) and its related problems have attracted wide attentions due to their wide applications in many disciplines (see, for example, [1–7]). Among these studies, a large number of research works have been devoted to developing efficient and implementable algorithms to solve VI(1.1) and its related problems (see, for example, [3–16]). In 1970, Sibony [7] proposed the classical gradient projection algorithm: xi+1 = PC (xi − λF (xi )) ,
(1.2)
where F is strongly monotone and L-Li
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