Hybrid viscosity approach for hierarchical fixed-point problems

Abstract

In this paper, we propose hybrid implicit and explicit viscosity iterative algorithms for solving general hierarchical fixed-point problems for a countable family of non-expansive mappings in uniformly smooth Banach spaces. These hybrid viscosity algorithms are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under suitable conditions. Our results extend, improve, supplement and develop the recent results in the literature.

Keywords:

• hybrid viscosity iterative algorithms
• non-expansive mappings
• uniformly smooth Banach spaces
• hierarchical fixed-point problems
• Demiclosedness principle.

AMS Subject Classifications:

• 49J30
• 47H09
• 47J20

Acknowledgments

The research part of first and second author was done during the visit to second author to Tabuk University, Tabuk, Saudi Arabia and it was supported by a research grant no. S1434-0119 from Tabuk University, Tabuk, Saudi Arabia. Third author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and Ph.D. Program Foundation of Ministry of Education of China (20123127110002).

Notes

Dedicated to Professor Boris S. Mordukhovich on the occasion of his 65th birthday.