Monotone Vector Fields and Generation of Nonexpansive Semigroups in Complete CAT(0) Spaces

Abstract

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in $\text{CAT}(0)$ spaces stands in opposed to the monotonicity defined earlier in $\text{CAT}(0)$ spaces by Khatibzadeh and Ranjbar [J. Aust. Math. Soc. 103(1), 70–90 (2017).] and Chaipunya and Kumam [Optimization 66(10), 1647–1665 (2017).]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida approximations of a given monotone vector field and derive many of their important properties. Finally, we prove a generation theorem by showing convergence of an exponential formula applied to resolvents of a monotone vector field. Our findings improve several known results in the literature including generation theorems of Jost [AMS/IP Stud. Adv. Math., vol. 8, pp. 1–47. Amer. Math. Soc., Providence, RI (1998)], Mayer [Comm. Anal. Geom. 6(2), 199–253 (1998).], Stojkovic [Adv. Calc. Var. 5(1), 77–126 (2012).], and Bačák [Convex analysis and optimization in Hadamard spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 22. De Gruyter, Berlin (2014).] for proper, convex, lower semicontinuous functions in the context of complete $\text{CAT}(0)$ spaces, and also by Iwamiya and Okochi [Nonlinear Anal. 54(2), 205–214 (2003).] for monotone vector fields in the context of Hadamard manifolds.

Keywords:

• monotone vector field
• nonexpansive semigroup
• resolvent
• space
• tangent space
• Yosida approximation

2010 MATHEMATICS SUBJECT CLASSIFICATION::

• 90C33
• 65K15
• 49J40
• 49M30
• 47H05

Acknowledgements

The authors are grateful to the reviewers for their helpful comments which have improved the manuscript magnificently.

Additional information

Funding

This research is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2021 under project number 64A306000005. The second author was supported by JSPS KAKENHI Grant No. 17K05372.