# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021289
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## The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times

 School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China

* Corresponding author: Yunfei Peng

Received  August 2021 Early access December 2021

Fund Project: The first author is supported by the Foundation of Postgraduate of Guizhou Province grant 2019032; The second author is supported by the National Natural Science Foundation of China grant 12061021 and 11661020

In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching $k\in \mathbb{N}$ times on bounded time intervals.

Citation: Huanting Li, Yunfei Peng, Kuilin Wu. The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021289
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