Stability Analysis For Two Classes Of Impulsive Integro-diferential Systems

In this paper, we consider the stability properties of the following impulsive integro-diferential equations with fixed time impul sesand one class of delay impuIn system（I）,f（t,x,Tx）=F（t,x）+R（t,x,Tx）,Tx=f_t0^tL（t,s,x,（s））ds,L∈C（R₊²xRⁿ,Rⁿ）.In system（IV）,Tx=_t-7^tk（s,x,（s））ds,t＞0，Ik∈C（R,R）,F∈C（t0,∞）xRxR,R),k∈C（R⁺xR,R）,h∈C（[t₀,∞）xR,R),we mainly study their stabilities.Impulsive integro-diferential system has extensive practical background and ap-plication in the natural science,as the systems extensively occur in the mathematicalmodeling of circuit simulation in physics and neuronal networks in biology.The researchof impulsive integro-diferential system has aroused experts’interest and attention,andvarious interesting results have been obtained in the past years[1 11,18]. In the process ofthe study of this system article[2] got the comparison criteria of stability of the trivialsolution of impulsive integro-diferential syetems,and article[1,3,4]studied boundednessof solutions of this system and got some direct results.Howerver,the study of stabil-ity of this system is in underway phase,and there are many problems which are notsolved.Therefore,we have a large number of work to do.In this paper we study the prop-erties of stability of two classes of impulsive integro-diferential systems and we get somenew results.In the nature,many things’variance not only relied on conditions then,but also re-lied on conditions for its past time,namely there is a lag between reaction and the reasonwhich caused it,and often with impulsive diferential systems.There were many achieve-ments for its study in the recent ten years[18 19,27 32,38 45],and23stability method is very useful[38 45].the research of impulsive integro-diferential system has aroused experts’in-terest and attention,and has made important progress in recent years,but the researchof delay impulsive integro-diferential system is quite scarce[46 49], so there are multiplyworks to do.In chapter one,using the method of variational Lyapunov functions and comparisonprinciple,we discuss the property of impulsive integro-diferential system with fixed timeimpulses. Diferent from the past,combined with Razumikhin technology,we can estab-lish comparison principle whose condition is easier to be satisfied.we firstly give somerelated preparation knowledge of impulsive integro-diferential system with fixed timeimpulses,then we establish a new comparison principle combined with Razumikhin tech-nology by comparing with the impulsive diferential system.In corollaries,by taking therelevant function of comparison theorem,the solution of the perturbed system （I） andunperturbed one are connected through the maximal solutions of the comparison sys-tem.Then on the base of comparison theorem,we get the comparison criteria of stabilityin terms of two measures of system （I）.At the same time,an example is given to showthe efectiveness of the theorems.In chapter two,on the base of comparison theorem in the first chapter,we get somecomparison criteria of the practical stability,the strict stability and the eventually sta-bility in terms of two measures of system （I） .At the same time,an example is given toshow the efectiveness of the theorems.In chapter three we give some related preparation knowledge of the delay impulsiveintegro-diferential system（IV ） first,then investigate its stability and get some criterionsfor uniformly stability and asymptotic stability.An example is given to show the efec-tiveness of the theorems.