Strong convergence theorems for maximal monotone operators in Banach spaces

Abstract

Let E be a uniformly convex and uniformly smooth Banach space with the dual E* and let T : E → 2 ^E* be a maximal monotone operator. By using the technique of resolvent operators and by using modified Ishikawa iteration and modified Halpern iteration for relatively non-expansive mappings, we suggest and analyse two iterative algorithms for finding an element x ∈ E such that 0 ∈ T(x). Strong convergence theorems for such iterative algorithms are proved. The ideas of these algorithms are applied to solve the problem of finding a minimizer of a convex function on E.

Keywords:

• maximal monotone operator
• uniformly convex and uniformly smooth Banach space
• generalized projection
• technique of resolvent operators
• iterative algorithms
• strong convergence

AMS Subject Classifications::

• 47H05
• 47J25
• 47H09

Acknowledgements

In this research, L.C. Ceng was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), National Science Foundation of China (10771141), PhD Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Shanghai Leading Academic Discipline Project (S30405), and Innovation Program of Shanghai Municipal Education Commission (0922133). J.C. Yao was partially supported by the grant NSC 96-2119-M-110-001.