Mann-Type Steepest-Descent and Modified Hybrid Steepest-Descent Methods for Variational Inequalities in Banach Spaces

Abstract

In this paper, we propose three different kinds of iteration schemes to compute the approximate solutions of variational inequalities in the setting of Banach spaces. First, we suggest Mann-type steepest-descent iterative algorithm, which is based on two well-known methods: Mann iterative method and steepest-descent method. Second, we introduce modified hybrid steepest-descent iterative algorithm. Third, we propose modified hybrid steepest-descent iterative algorithm by using the resolvent operator. For the first two cases, we prove the convergence of sequences generated by the proposed algorithms to a solution of a variational inequality in the setting of Banach spaces. For the third case, we prove the convergence of the iterative sequence generated by the proposed algorithm to a zero of an operator, which is also a solution of a variational inequality.

Keywords:

• Convergence analysis
• Mann-type steepest-descent method
• Modified hybrid steepest-descent method
• Nonexpansive maps
• Resolvent operators
• Variational inequalities

AMS Subject Classification:

• 49J40
• 47J20
• 47H10
• 47H06
• 65K10

ACKNOWLEDGMENTS

In this research, the first author was partially supported by the National Science Foundation of China (10771141), the Ph.D. Program Foundation of the Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). The second author was supported by a KFUPM Funded Research Project (IN070362), and the third author was partially supported by grant NSC97-2115-M-110-001 from the National Science Council of Taiwan.