scholarly journals Some Algorithms for Finding Fixed Points and Solutions of Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Jong Soo Jung
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Minimum Norm ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mapping ◽  
Unique Minimum ◽  
Implicit And Explicit

We introduce new implicit and explicit algorithms for finding the fixed point of ak-strictly pseudocontractive mapping and for solving variational inequalities related to the Lipschitzian and strongly monotone operator in Hilbert spaces. We establish results on the strong convergence of the sequences generated by the proposed algorithms to a fixed point of ak-strictly pseudocontractive mapping. Such a point is also a solution of a variational inequality defined on the set of fixed points. As direct consequences, we obtain the unique minimum-norm fixed point of ak-strictly pseudocontractive mapping.

scholarly journals Computing the Fixed Points of Strictly Pseudocontractive Mappings by the Implicit and Explicit Iterations

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Yeong-Cheng Liou
Keyword(s):  
Fixed Point ◽  
Inverse Problems ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Minimum Norm ◽  
Pseudocontractive Mappings ◽  
Special Cases ◽  
Implicit And Explicit ◽  
Strictly Pseudocontractive Mappings

It is known that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings have been constructed in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

scholarly journals An Iterative Algorithm for Solving Generalized Variational Inequalities and Fixed Points Problems

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 61 ◽  
Author(s):  
Yonghong Yao ◽  
Mihai Postolache ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonlinear Transformation ◽  
Generalized Variational Inequalities ◽  
Generalized Variational Inequality ◽  
Suggested Algorithm

In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested algorithm is demonstrated.

scholarly journals Extragradient algorithms for variational inequality, fixed point and generalized mixed equilibrium problems

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1021-1030
Author(s):  
Baohua Guo ◽  
Lijuan Sun
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The purpose of this paper is to investigate variational inequalities, fixed point problems and generalized mixed equilibrium problems. Anextragradient iterative algorithm is investigated in the framework of Hilbert spaces. Weak convergence theorems for common solutions are established.

scholarly journals Strong convergence theorems for variational inequalities and fixed point problems in Banach spaces

2021 ◽  
Vol 37 (3) ◽  
pp. 477-487
Author(s):  
MONDAY OGUDU NNAKWE ◽  
◽  
" JERRY N." EZEORA ◽  
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Metric Projection ◽  
Finite Family ◽  
Common Solution ◽  
Monotone Type

In this paper, using a sunny generalized non-expansive retraction which is different from the metric projection and generalized metric projection in Banach spaces, we present a retractive iterative algorithm of Krasnosel’skii-type, whose sequence approximates a common solution of a mono-variational inequality of a finite family of η-strongly-pseudo-monotone-type maps and fixed points of a countable family of generalized non-expansive-type maps. Furthermore, some new results relevant to the study are also presented. Finally, the theorem proved complements, improves and extends some important related recent results in the literature.

scholarly journals IMPLICIT AND EXPLICIT ALGORITHMS FOR MINIMUM-NORM FIXED POINTS OF PSEUDOCONTRACTIONS IN HILBERT SPACES

2012 ◽  
Vol 16 (4) ◽  
pp. 1489-1506 ◽  
Author(s):  
Yonghong Yao ◽  
Vittorio Colao ◽  
Giuseppe Marino ◽  
Hong-Kun Xu
Keyword(s):  
Fixed Points ◽  
Hilbert Spaces ◽  
Minimum Norm ◽  
Implicit And Explicit

scholarly journals Viscosity Projection Algorithms for Pseudocontractive Mappings in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xiujuan Pan ◽  
Shin Min Kang ◽  
Young Chel Kwun
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Projection Algorithm ◽  
Strong Convergence Theorem ◽  
Minimum Norm ◽  
Projection Algorithms ◽  
Pseudocontractive Mapping ◽  
Pseudocontractive Mappings

An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping.

scholarly journals Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 142 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
General System ◽  
Inequality Constraint ◽  
Common Fixed Points ◽  
Common Solution ◽  
Implicit Iteration

In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guaranteed under some suitable assumptions. As a sub-result, we provide an algorithm to solve problem of common fixed points of pseudocontractive, nonexpansive mappings, variational inequality problems and generalized mixed bifunction equilibrium problems in Hilbert spaces.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

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