Hybrid Steepest Descent Method for Variational Inequality Problem over the Fixed Point Set of Certain Quasi-nonexpansive Mappings

Yamada, Isao; Ogura, Nobuhiko

doi:10.1081/nfa-200045815

Cited by 225 publications

(161 citation statements)

References 41 publications

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“…[7,8,11,12,13,14]) studied the viscosity approximation method as follow: for given x 0 ∈ C, the sequence {x n } is generated by…”

Section: Given X Y ∈ H Letmentioning

confidence: 99%

A viscosity projection method for class T mappings

Dong

2013

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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In this paper, we firstly introduce a viscosity projection method for the class T mappingswhere Sn = (1 − w)I + wTn, w ∈ (0, 1), Tn ∈ T and prove strong convergence theorems of the proposed method. It is verified that the viscosity projection method converges locally faster than the viscosity method. Furthermore, we present a viscosity projection method for a quasi-nonexpansive and nonexpansive mappings in Hilbert spaces. A numerical test provided in the paper shows that the viscosity projection method converges faster than the viscosity method.

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“…[7,8,11,12,13,14]) studied the viscosity approximation method as follow: for given x 0 ∈ C, the sequence {x n } is generated by…”

Section: Given X Y ∈ H Letmentioning

confidence: 99%

A viscosity projection method for class T mappings

Dong

2013

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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“…As a matter of fact, the above algorithms (1.1) and (1.2) have only weak convergence except in a finite dimensional space. To obtain strong convergence in the setting of an infinite dimensional Hilbert or Banach spaces, there exist several iterative algorithms to nonexpansive mappings (e.g., Viscosity iteration algorithm [7], Hybrid projection algorithm [8], Hybrid steepest descent algorithm [9], Halpern-type iteration algorithm [10,11], Shrinking projection algorithm [12], etc.). In general, the nonexpansive mapping may have more than one fixed point.…”

Section: Introductionmentioning

confidence: 99%

Iterative algorithms for minimum-norm fixed point of non-expansive mapping in hilbert space

Cai

Tang

Liu

2012

Fixed Point Theory Appl

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The purpose of this article is to introduce two iterative algorithms for finding the least norm fixed point of nonexpansive mappings. We provide two algorithms, one implicit and another explicit, from which strong convergence theorems are obtained in Hilbert spaces. Then we apply these algorithms to solve some convex optimization problems. Furthermore, we use them to solve some split feasibility problems. The results of this article extend and improve several results presented in the literature in the recent past. Mathematics Subject Classification (2000): 47H09; 47H10; 90C25.

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“…Yamada and Ogura [YO04] and Mainge [Mai10] named the cutters firmly quasinonexpansive operators. These operators were named directed operators in Zaknoon [Zak03] and further employed under this name by Segal [Seg08] and Censor and Segal [CS08,CS08a,CS09].…”

Section: Preliminariesmentioning

confidence: 99%

Extrapolation and local acceleration of an iterative process for common fixed point problems

Cegielski¹,

Censor²

2012

Journal of Mathematical Analysis and Applications

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We consider sequential iterative processes for the common fixed point problem of families of cutter operators on a Hilbert space. These are operators that have the property that, for any point x ∈ H, the hyperplane through T x whose normal is x − T x always "cuts" the space into two half-spaces one of which contains the point x while the other contains the (assumed nonempty) fixed point set of T. We define and study generalized relaxations and extrapolation of cutter operators and construct extrapolated cyclic cutter operators. In this framework we investigate the Dos Santos local acceleration method in a unified manner and adopt it to a composition of cutters. For these we conduct convergence analysis of successive iteration algorithms.

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Hybrid Steepest Descent Method for Variational Inequality Problem over the Fixed Point Set of Certain Quasi-nonexpansive Mappings

Cited by 225 publications

References 41 publications

A viscosity projection method for class T mappings

A viscosity projection method for class T mappings

Iterative algorithms for minimum-norm fixed point of non-expansive mapping in hilbert space

Extrapolation and local acceleration of an iterative process for common fixed point problems

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