Abstract
In this paper we introduce a multi-step implicit iterative scheme with regularization for finding a common solution of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces. The multi-step implicit iterative method with regularization is based on three well-known methods: the extragradient method, approximate proximal method and gradient projection algorithm with regularization. We derive a weak convergence theorem for the sequences generated by the proposed scheme. On the other hand, we also establish a strong convergence result via an implicit hybrid method with regularization for solving these two problems. This implicit hybrid method with regularization is based on the CQ method, extragradient method and gradient projection algorithm with regularization.
| Original language | English |
|---|---|
| Article number | 240 |
| Journal | Journal of Inequalities and Applications |
| Volume | 2013 |
| DOIs | |
| State | Published - Dec 2013 |
Bibliographical note
Funding Information:The first 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). The last author was partially supported by the a grant from the NSC 101-2115-M-037-001.
Keywords
- Implicit hybrid method with regularization
- Inverse-strong monotonicity
- Kadec-Klee property
- Minimization problem
- Multi-step implicit iterative method with regularization
- Nonexpansive mapping
- Opial's condition
ASJC Scopus subject areas
- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics