Existences And Asymptotic Properties Of The Solutions Of Non-instantaneous Impulsive Differential Equations

Non-instantaneous impulsive differential equation is a generalization of the classical instantaneous impulsive differential equation.Its characteristic is that the time of the impulsive action is not negligible with respect to the entire development process.Recent years,non-instantaneous impulsive differential equations have received extensive attention from many researchers,the theoretical results have many applications in pest control and prevention,pharmacokinetics,and other aspects.The main work of the article is divided into two parts.Firstly,we study the existence of solutions of integral boundary value problems for non-instantaneous impulsive differential equations.We convert the problem of the existence of solutions to the integerorder integral boundary value problem to the fixed-point problem of the corresponding operator.Combining the principle of contraction mapping and the Krasnoselskii fixed point theorem,the existence and uniqueness of the solutions are obtained.Then use the same ideal to analyze and study the fractional situation.Secondly,we study the asymptotic properties of nonlinear non-instantaneous impulsive differential equations in infinite time interval.It mainly study the continuous dependence and stability of the solutions of the equations.We give the definition of the continuous dependence and stability of the solutions to integer-order non-instantaneous impulsive differential equations,and classify the position distribution of the impulsive points and the junction points on the positive half time axis,use the pulsed Gronwall inequality and other techniques to obtain the new results for continuous dependence and stability of the solutions.In addition,we introduce a new concept of fractional gravitational constant to describe the stability of the solution of fractional-order problem,then extend the integer order results to the fractional situation.