On a Two-Step Algorithm for Hierarchical Fixed Point Problems and Variational Inequalities

Cianciaruso, Filomena; Marino, Giuseppe; Muglia, Luigi; Yao, Yonghong

doi:10.1155/2009/208692

Cited by 37 publications

(20 citation statements)

References 7 publications

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“…5 Center for General Education, China Medical University, Taichung, 40402, Taiwan. 6 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.…”

Section: Competing Interestsmentioning

confidence: 99%

An iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings

Bnouhachem¹,

Ansari²,

Yao³

2015

Fixed Point Theory Appl

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This paper aims to deal with an iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings in the setting of real Hilbert spaces. We establish the strong convergence of the proposed method under some suitable conditions. Numerical examples are presented to illustrate the proposed method and convergence result. The algorithm and result presented in this paper extend and improve some well-known algorithm and results, respectively, in the literature.MSC: Primary 49J30; 47H09; secondary 47J20; 49J40

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“…5 Center for General Education, China Medical University, Taichung, 40402, Taiwan. 6 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.…”

Section: Competing Interestsmentioning

confidence: 99%

An iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings

Bnouhachem¹,

Ansari²,

Yao³

2015

Fixed Point Theory Appl

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“…It is known that the hierarchical fixed point problem (1) links with some monotone variational inequalities and convex programming problems; see [3,4,5,6,7,8] In 2006, Marino and Xu [16] introduced the viscosity iterative method for nonexpansive mappings. They considered the following general iterative method:…”

Section: Introductionmentioning

confidence: 99%

Strong convergence with a modified iterative projection method for hierarchical fixed point problems and variational inequalities

Karahan¹,

Özdemir²

2016

ntmsci

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Let C be a nonempty closed convex subset of a real Hilbert space H. Let {T n } : C → H be a sequence of nearly nonexpansive mappings such thatC → H be a γ-Lipschitzian mapping and F : C → H be a L-Lipschitzian and η-strongly monotone operator. This paper deals with a modified iterative projection method for approximating a solution of the hierarchical fixed point problem. It is shown that under certain approximate assumptions on the operators and parameters, the modified iterative sequence {x n } converges strongly to x * ∈ F which is also the unique solution of the following variational inequality:As a special case, this projection method can be used to find the minimum norm solution of above variational inequality; namely, the unique solution x * to the quadratic minimization problem: x * = ar g mi n x∈F x 2 . The results here improve and extend some recent corresponding results of other authors.

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“…Korpelevich's extragradient method has extensively been studied in the literature for solving a more general problem that consists of finding a common point that lies in the solution set of a variational inequality and the set of fixed points of a nonexpansive mapping. This type of problem aries in various theoretical and modeling contexts, see e.g., [2], [4]- [7], [15], [25], [26] and references therein. Especially, Nadezhkina and Takahashi [17] introduced the following iterative method which combines Korpelevich's extragradient method and a CQ method:…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of a hybrid method for pseudomonotone variational inequalities and fixed point problems

Yao

Liou

2012

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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In this paper, we suggest a hybrid method for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. We derive a necessary and sufficient condition for the strong convergence of the sequences generated by the proposed method.

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On a Two-Step Algorithm for Hierarchical Fixed Point Problems and Variational Inequalities

Cited by 37 publications

References 7 publications

An iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings

An iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings

Strong convergence with a modified iterative projection method for hierarchical fixed point problems and variational inequalities

Strong convergence of a hybrid method for pseudomonotone variational inequalities and fixed point problems

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