Proximal point methods with possible unbounded errors for monotone operators in Hadamard spaces

Abstract

In this paper, we investigate and analyse the strong convergence of the sequence generated by an inexact proximal point method with possible unbounded errors for finding zeros of monotone operators in Hadamard spaces. We show that the boundedness of the generated sequence is equivalent to the zero set of the operator to be nonempty. In this case, we prove the strong convergence of the generated sequence to a zero of the operator. We also provide some applications of our main results and give a numerical example to show the performance of the proposed algorithm.

Keywords:

• Monotone operator
• proximal point method
• resolvent
• strong convergence
• unbounded error

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 47H05
• 47J25
• 47H10
• 47H09
• 47H04

Acknowledgments

The authors are grateful to the Editor and the anonymous referees for their constructive comments leading to the improvement of the paper. This work was done while the second author was visiting the University of Texas at El Paso. The second author would like to thank Professor Djafari Rouhani and the Department of Mathematical Sciences for their kind hospitality at the University of Texas at El Paso during his visit.

Disclosure statement

No potential conflict of interest was reported by the author(s).