Perturbed iterative methods for a general family of operators: convergence theory and applications

ABSTRACT

We study perturbations of a hybrid steepest descent method for locating common fixed points of an arbitrary pool $\left\{T_{\lambda }\right\}$ of nonexpansive mappings. The difficulty of handling a possibly uncountable family is settled down to dealing with a countable family of auxiliary mappings $\left\{S_n\right\}$ associated to $\left\{T_{\lambda }\right\}$, in the sense that the approximate fixed points of $\left\{S_n\right\}$ provide the common fixed points of $\left\{T_{\lambda }\right\}$. Algorithms with strong convergence for solving the associated variational inequality problems are presented. Applications to convex minimization problems and convex feasibility problems are provided, together with numerical examples for comparisons of our algorithms with the existing ones.

Keywords:

• Convex feasibility problems
• convex minimization problems
• firmly nonexpansive mappings
• multiple-set split feasibility problems
• steepest descent-like methods
• variational inequality problems

2000 Mathematics Subject Classifications:

• 47H09
• 47H10
• 47J25

Acknowledgments

We would like to express our deep gratitude to the referees for many useful ideas and suggestions to improve the paper.

This work is supported by Taiwan MOST grants 105-2115-M-039-002-MY3 (for Yao) and 108-2115-M-110-004-MY2 (for Wong).

Authors' contributions

All authors contribute equally and significantly in writing this paper. All readers read and approved the final manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by Taiwan MOST grants 105-2115-M-039-002-MY3 (for Yao) and 108-2115-M-110-004-MY2 (for Wong).