Noncoercive convex sweeping processes with velocity constraints

Abstract

In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result for coercive convex sweeping processes with velocity constraints and a useful fact on measurable single-valued mappings, we prove the solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable conditions. These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work [Adly et al., Convex and nonconvex sweeping processes with velocity constraints: well-posedness and insights. Appl Math Optim. 2023;88(Paper No. 45)]. Additionally, we resolve two other open questions from the same paper concerning the behaviour of the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.

Keywords:

• Sweeping process
• velocity constraint
• monotone operator
• parametric variational inequality
• the Tikhonov regularization

2020 Mathematics Subject Classifications:

• 47H05
• 46N10

Acknowledgments

The research of Samir Adly was funded by Institut de recherche XLIM, Université de Limoges. Research of Nguyen Nang Thieu was supported by the project DLTE00.01/22-23 of Vietnam Academy of Science and Technology. Nguyen Dong Yen was supported by Institute of Mathematics, Vietnam Academy of Science and Technology, the XLIM laboratory at the University of Limoges, and the Margot research federation. The authors highly appreciate the careful review of the paper and the valuable suggestions of the two anonymous referees.

Disclosure statement

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

Notes

1 In the whole paper [9], it suffices to assume that the set-valued mapping $C:[0,T]\rightrightarrows \mathcal{H}$ is Lipschitz-like around every point $(t,x)$ in its graph with $t\in (0,T)$. Indeed, in the proof of [9, Theorem 3.7], by taking $\overline{\lambda }=\overline{t}\in (0,T)$ one can show that the formula $z\left(t\right)=x\left(\mu \right(t),t)$ defines a continuous function $z:(0,T)\to \mathcal{H}$. Then, with a slight modification, the remaining of the proof of [9, Theorem 3.7] goes well as it is.