Convergence results for a zero of the sum of a finite family of maximal monotone mappings in Banach spaces

ABSTRACT

The purpose of this paper is to study the method of approximation for zeros of the sum of a finite family of maximal monotone mappings in the setting of Banach spaces. Under some mild conditions, we establish strong convergence results of the proposed approximation method. The assumptions that one of the mappings is single valued and α-inverse strongly monotone are dispensed with. In addition, we give some applications to the minimization problems. Finally, we provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

Keywords:

• Firmly nonexpansive
• Hilbert spaces
• maximal monotone mapping
• strong convergence
• zero points

2010 Mathematics Subject Classifications:

• 47H05
• 47H09
• 47H10
• 47J25
• 90C25

Acknowledgements

The authors express their deep gratitude to the referee and the editor for his/her valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first and the second authors gratefully acknowledge the funding received from Simons Foundation based at Botswana International University of Science and Technology (BIUST) [ID267269FY17].