Revisiting jointly firmly nonexpansive families of mappings

Abstract

Recently, the author, together with L. Leuştean and A. Nicolae, introduced the notion of jointly firmly nonexpansive families of mappings in order to investigate in an abstract manner the convergence of proximal methods. Here, we further the study of this concept, by giving a characterization in terms of the classical resolvent identity, by improving on the rate of convergence previously obtained for the uniform case, and by giving a treatment of the asymptotic behaviour at infinity of such families.

Keywords:

• Convergence of resolvents
• proximal point algorithm
• CAT(0) spaces
• jointly firmly nonexpansive families
• proof mining

Mathematics Subject Classification 2010:

• 90C25
• 46N10
• 47J25
• 47H09
• 03F10

Acknowledgments

This work has been supported by the German Science Foundation (DFG Project KO 1737/6-1) and by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PN-III-P1-1.1-PD-2019-0396. I would like to thank Liviu Păunescu for suggesting to investigate continuity in Theorem 5.3.

Disclosure statement

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

Additional information

Funding

This work has been supported by the German Science Foundation (Deutsche Forschungsgemeinschaft) (DFG Project KO 1737/6-1) and by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, (Ministerul Educaţiei şi Cercetării Ştiinţifice), project number PN-III-P1-1.1-PD-2019-0396.