Hybrid Iterative Scheme for Triple Hierarchical Variational Inequalities with Mixed Equilibrium, Variational Inclusion, and Minimization Constraints

We introduce and analyze a hybrid iterative algorithm by combining Korpelevich's extragradient method, the hybrid steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.

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On the triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions and systems of generalized equilibria

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.45.2014.1658 ◽

2014 ◽

Vol 45 (3) ◽

pp. 297-334 ◽

Cited By ~ 5

Author(s):

Jen-Chih Yao ◽

Cheng Lu-chuan

Keyword(s):

Variational Inequality ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Extragradient Method ◽

Solution Set ◽

Steepest Descent Method ◽

Generalized Mixed Equilibrium Problem ◽

Common Element ◽

Variational Inclusions ◽

Mixed Equilibria

In this paper, we introduce and analyze a relaxed iterative algorithm by combining Korpelevich's extragradient method, hybrid steepest-descent method and Mann's iteration method. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions and the solution set of a system of generalized equilibrium problems (SGEP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the SGEP and finitely many variational inclusions.

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Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

Journal of Function Spaces ◽

10.1155/2015/980352 ◽

2015 ◽

Vol 2015 ◽

pp. 1-22

Author(s):

L. C. Ceng ◽

A. Latif ◽

C. F. Wen ◽

A. E. Al-Mazrooei

Keyword(s):

Variational Inequality ◽

Variational Inequalities ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Extragradient Method ◽

Steepest Descent ◽

Solution Set ◽

Steepest Descent Method ◽

Common Element ◽

Variational Inclusions

We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. The results obtained in this paper improve and extend the corresponding results announced by many others.

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The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

Journal of Applied Mathematics ◽

10.1155/2012/282094 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17

Author(s):

Rabian Wangkeeree ◽

Panatda Boonman

Keyword(s):

Hilbert Space ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

Solution Set ◽

Common Element ◽

Fixed Point Set ◽

Mixed Equilibrium Problems ◽

Point Set ◽

Mixed Equilibrium

We first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptoticTJmapping and the solution set of SMEP in a real Hilbert space.

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Strong Convergence Theorems for Family of Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/734082 ◽

2011 ◽

Vol 2011 ◽

pp. 1-22

Author(s):

Yekini Shehu

Keyword(s):

Variational Inequality ◽

Strong Convergence ◽

Equilibrium Problems ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Common Element ◽

Generalized Mixed Equilibrium Problems ◽

Mixed Equilibrium Problems ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of nonexpansive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three sets. We give some applications of our results. Our results extend important recent results.

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Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

Journal of Applied Mathematics ◽

10.1155/2012/538912 ◽

2012 ◽

Vol 2012 ◽

pp. 1-29 ◽

Cited By ~ 2

Author(s):

Tanom Chamnarnpan ◽

Poom Kumam

Keyword(s):

Fixed Point ◽

Equilibrium Problems ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Monotone Mapping ◽

Iterative Algorithms ◽

Infinite Family ◽

Common Element ◽

Mixed Equilibrium Problems ◽

Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

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A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

Abstract and Applied Analysis ◽

10.1155/2010/123027 ◽

2010 ◽

Vol 2010 ◽

pp. 1-31 ◽

Cited By ~ 15

Author(s):

Siwaporn Saewan ◽

Poom Kumam

Keyword(s):

Banach Space ◽

Variational Inequality ◽

Monotone Operator ◽

Maximal Monotone Operator ◽

Iterative Scheme ◽

Nonexpansive Mappings ◽

Maximal Monotone ◽

Generalized Mixed Equilibrium Problem ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

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Projection Algorithms for Variational Inclusions

Journal of Applied Mathematics ◽

10.1155/2012/137932 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11

Author(s):

Youli Yu ◽

Pei-Xia Yang ◽

Khalida Inayat Noor

Keyword(s):

Hilbert Space ◽

Variational Inequality ◽

Iterative Algorithm ◽

Real Hilbert Space ◽

Projection Algorithm ◽

Variational Inclusion ◽

Inclusion Problem ◽

Variational Inclusions ◽

Projection Algorithms ◽

Variational Inclusion Problem

We present a projection algorithm for finding a solution of a variational inclusion problem in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a solution of the variational inclusion problem which also solves some variational inequality.

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Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces

Abstract and Applied Analysis ◽

10.1155/2014/271208 ◽

2014 ◽

Vol 2014 ◽

pp. 1-17

Author(s):

Poom Kumam ◽

Thanyarat Jitpeera

Keyword(s):

Iterative Algorithm ◽

Equilibrium Problems ◽

Real Hilbert Space ◽

Nonexpansive Mappings ◽

Infinite Family ◽

Variational Inclusion ◽

Common Element ◽

Point Problem ◽

Mixed Equilibrium Problems ◽

Mixed Equilibrium

We introduce a new iterative algorithm for approximating a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve the results of Peng and Yao, 2009, Qin et al. 2010 and many authors.

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A New Hybrid Algorithm for a Pair of Quasi--Asymptotically Nonexpansive Mappings and Generalized Mixed Equilibrium Problems in Banach Spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/956852 ◽

2011 ◽

Vol 2011 ◽

pp. 1-22

Author(s):

Jinhua Zhu ◽

Shih-Sen Chang

Keyword(s):

Strong Convergence ◽

Nonexpansive Mappings ◽

Convex Banach Space ◽

Generalized Mixed Equilibrium Problem ◽

Common Element ◽

Strong Convergence Theorem ◽

Asymptotically Nonexpansive Mappings ◽

Asymptotically Nonexpansive ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi--asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

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Hybrid Algorithms for Minimization Problems over the Solutions of Generalized Mixed Equilibrium and Variational Inclusion Problems

Mathematical Problems in Engineering ◽

10.1155/2011/648617 ◽

2011 ◽

Vol 2011 ◽

pp. 1-25 ◽

Cited By ~ 5

Author(s):

Thanyarat Jitpeera ◽

Poom Kumam

Keyword(s):

Real Hilbert Space ◽

Monotone Mapping ◽

Variational Inclusion ◽

Generalized Mixed Equilibrium Problem ◽

Common Element ◽

Inclusion Problems ◽

Minimization Problems ◽

Inverse Strongly Monotone ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

We introduce a new general hybrid iterative algorithm for finding a common element of the set of solution of fixed point for a nonexpansive mapping, the set of solution of generalized mixed equilibrium problem, and the set of solution of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (2006), Yao and Liou (2010), Tan and Chang (2011), and other authors.

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