Several Class Of Non-Instantaneous Impulsive Differential Equations With Initial And Boundary Value Problem

Impulsive differential equation,as an important branch of differential equation,has become one of the research hotspots in recent years.It has been widely used in the fields of physics,chemistry,population dynamics,biology and economics,and the development of this theory is becoming more and more mature.In 2013,Hernadez and O'egan proposed a new class of impulsive differential equations.The difference between this equation and the previous impulsive differential equations is that after the state suddenly changes,it will last for a limited period of time.The background of this equation is to describe the process of insulin absorption in diabetic patients after insulin injection.This equation is then referred to as the non-instantaneous impulsive differential equation.Form as was called non-instantaneous impulsive differential equations,where tk<sk+1.When sk+1=tk,the above non-instantaneous impulsive differential equations degenerate into instantaneous impulsive differential equations.This paper is divided into three chapters,and mainly discusses the existence of the solutions of several class of non-instantaneous impulsive differential equations with initial and boundary value problem.In Chapter 1,we consider initial value problem for first order non-instantaneous impul-sive differential equations here x0?R,p?A,0=s0?t0?s1?t1?s2?t2?s3?…?sp?tp?sp+1=T,fk?C([sk,tk]×R,R)and ?k?C(tk,sk+1]×R×R,R),k=0,1,…,p.By establishing new comparison results,the existence of solutions to the above problems is obtained with the upper and lower solutions method and monotone iterative technique.Our results generalize and improve the existing results in the corresponding literatures.Finally,an example is given to illustrate the improvement of the results.In Chapter 2,we consider first order non-instantaneous impulsive differential equations with boundary value problems where fk?C((sk,tk+1]×R,R),k=0,1,…,p,?k?C((tk,sk]×R×R,R),k=1,2,…,p,0=s0<t1?s1?t2?s2?…?tp?sp?tp+1=T,p?Z,l>0.By using method of upper and lower solutions coupled with monotone iterative technique,existence results is obtained.And an example is used to verify the rationality of the existence of the theorem.In Chapter 3,we consider first order non-instantaneous delay impulsive differential equations with nonlinear boundary condition where fk?C((sk,tk]×R×R,R),Lk?C((tk,sk+1),R),k=0,1,…,p,g?C(R x R,R),0=s0<t0?s1?t1?s2?t2?s3?…?sp?tp?sp+1=T,p?Z,r>0,?>0 and T?R.By establishing the existence of the solution of the corresponding linear delay differential equation and the corresponding comparison result,the existence of the above-mentioned problem solution is obtained by combining the monotone iterative technique.