Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of an equilibrium problem and fixed point problems

Fixexd point theory and applications

Qamrul Hasan Ansari¹  Abdul Latif²  Lu-Chuan Ceng³ 

[1] Department of Mathematics, Aligarh Muslim University, Aligarh, India;Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia;Department of Mathematics, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai Normal University, Shanghai, China

关键词: multi-step hybrid viscosity method;    systems of variational inequalities;    equilibrium problems;    fixed point problems;    convergence analysis;   

DOI  :  10.1186/1687-1812-2012-186

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we consider a system of variational inequalities defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping. We also consider a triple hierarchical variational inequality problem, that is, a variational inequality problem defined over a set of solutions of another variational inequality problem which is defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping. These two problems are very general and include, as special cases, several problems studied in the literature. We propose a multi-step hybrid viscosity method to compute the approximate solutions of our system of variational inequalities and a triple hierarchical variational inequality problem. The convergence analysis of the sequences generated by the proposed method is also studied. In addition, the nontrivial examples of two systems are presented and our results are applied to these examples. MSC:49J40, 47H05, 47H19.

【 授权许可】

CC BY