Explicit iteration methods for a class of variational inequalities in Banach spaces

Buong, Nguyen; Phuong, Nguyen Thi Hong; Thủy, Nguyễn Thị Thu

doi:10.3103/s1066369x15100023

Cited by 5 publications

(3 citation statements)

References 7 publications

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“…. This together with (24) gives that lim k→∞ T i x k − x k = 0 for all i = 1, • • • , m. Therefore, we also get (10). Further, from (21) with p replaced by p * , it follows (17).…”

Section: Lemma 22 ([46]mentioning

confidence: 71%

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Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

Buong

2022

JIMO

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<p style='text-indent:20px;'>In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.</p>

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Section: Lemma 22 ([46]mentioning

confidence: 71%

“…for each i ∈ L there exist at least an integer t with 1 ≤ t ≤ q k and an index i k l (t ) ∈ L k t such that i = i k l (t ). Then, from ( 16) we obtain that lim k→∞ T i x k − x k = 0 for all i ∈ L. So, we obtain (10). Further, noting (11), the inequality…”

Section: Lemma 22 ([46]mentioning

confidence: 77%

Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

Buong

2022

JIMO

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“…Without this weak continuity, recently, in and , the author and Phuong suggested several implicit iterative and regularization methods when C is the set of common fixed points for an infinite family of nonexpansive mappings on E . Explicit iterative methods for the class of variational inequalities were considered in and .…”

Section: Introductionmentioning

confidence: 99%

Steepest‐descent proximal point algorithms for a class of variational inequalities in Banach spaces

Buong

2018

Mathematische Nachrichten

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In this paper, we present a new approach to the problem of finding a common zero for a system of m‐accretive mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We propose an implicit iteration method and two explicit ones, based on compositions of resolvents with the steepest‐descent method. We show that our results contain some iterative methods in literature as special cases. An extension of the Xu's regularization method for the proximal point algorithm from Hilbert spaces onto Banach ones under simple conditions of convergence and a new variant for the method of alternating resolvents are obtained. Numerical experiments are given to affirm efficiency of the methods.

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Convergence of a hybrid viscosity approximation method for finding zeros of m-accretive operators

2019

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Explicit iteration methods for a class of variational inequalities in Banach spaces

Cited by 5 publications

References 7 publications

Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

Steepest‐descent proximal point algorithms for a class of variational inequalities in Banach spaces

Convergence of a hybrid viscosity approximation method for finding zeros of m-accretive operators

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