Hybrid Algorithms of Common Solutions of Generalized Mixed Equilibrium Problems and the Common Variational Inequality Problems with Applications

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.

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System of generalized mixed equilibrium problems, variational inequality, and fixed point problems

Fixed Point Theory and Applications ◽

10.1186/s13663-016-0583-7 ◽

2016 ◽

Vol 2016 (1) ◽

Cited By ~ 1

Author(s):

Morad Ali Payvand ◽

Sedigheh Jahedi

Keyword(s):

Fixed Point ◽

Variational Inequality ◽

Equilibrium Problems ◽

Fixed Point Problems ◽

Generalized Mixed Equilibrium Problems ◽

Mixed Equilibrium Problems ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

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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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A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces

Abstract and Applied Analysis ◽

10.1155/2010/390972 ◽

2010 ◽

Vol 2010 ◽

pp. 1-39 ◽

Cited By ~ 4

Author(s):

Wanpen Chantarangsi ◽

Chaichana Jaiboon ◽

Poom Kumam

Keyword(s):

Variational Inequality ◽

Equilibrium Problems ◽

Descent Method ◽

Steepest Descent Method ◽

Generalized Mixed Equilibrium Problems ◽

Hybrid Steepest Descent Method ◽

Mixed Equilibrium Problems ◽

Relaxed Cocoercive Mapping ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.

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