THE EXISTENCE AND PROPERTIES OF THE SOLUTION OF A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS WITH SWITCHING AT VARIABLE TIMES

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 is an element of N times on bounded time intervals.

Automated summary

This paper deals with the qualitative theory for a class of nonlinear differential equations with switching at variable times, including the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters, and the stability of the system. The global exponential stability is discussed, with necessary and sufficient conditions for SSVT. The paper also investigates the continuous dependence and differentiability of the solution with respect to the initial state and the switching line.