Novel regularized dynamical systems for solving hierarchical fixed point problems

Abstract

In this paper, we study some Krasnoselskii-Mann type dynamical systems in solving fixed point problems. The first one can be regarded as a continuous version of the Krasnoselskii-Mann iterations. We prove that the solution of this dynamical system converges weakly to a fixed point of the involving mapping. Next, we focus our attention on a regularized Krasnoselskii-Mann type dynamical system. Besides proving existence and uniqueness of strong global solutions, we show that the generated trajectories converge strongly to a unique solution of a variational inequality over the fixed point set. Also, we provide a convergence rate analysis for the regularized dynamical system.

Keywords:

• Dynamical system
• regularized methods
• variational inequalities
• fixed point problems
• Krasnoselskii-Mann theorem

Acknowledgments

The author thanks the anonymous peer reviewers and the editor for their constructive comments which helped to improve the paper.

Disclosure statement

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

Availability of data and materials

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

Additional information

Funding

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2021.02.