Existence Of Positive Periodic Solutions For Impulsive Delay Differential Equations

Many actual problem development are characterized by the fact that they expe-rience a change of state abruptly at certain development stage. For convenience, inthese mathematical processes simulation, we will neglect the fast change continuousperiod and further assume that this process is completed through the instantaneoussudden change. This kind of instantaneous sudden change phenomenon is usuallycalled the impulse phenomenon. Impulsive phenomenon exists extensively in variousdomains of modern science and technology. Their mathematical model often can beformulated by the impulsive differential equation. The most prominent characteristicof the impulsive differential equation is considering the instantaneous sudden changephenomenon in affecting on the development state fully. This reflect the thing's de-velopment discipline much more deeply and accurately. For example, these aspectshave been applied widely in astronautics technical control system, communication, lifesciences, medicine, economic domain. Because of receiving the impulsive conditioneffect, the original unstable system can be stead and the original system which hasno periodic solution may have periodic solutions. Especially, under the coexistenceof "impulse" and "delay", it is much more duplicated that whether differential equa-tion exists periodic solution or not. These all make the corresponding fundamentalresearch related to the impulsive delay differential equation extremely important. Thispaper divides into two chapters. We mainly consider the existence of positive periodicsolutions for several classes of impulsive delay differential equation.Chapter 1 deals with the following linear impulsive delay differential equationWhere a: [0,∞)→[0,∞),Ï„_i:[0,∞)→[0,∞), (i=0,1,…,n), p(t):[0,∞)→[0,∞) are local summableω-periodic function, f∈C([0,∞)ⁿ⁺², [0,∞)), and (?)(t,u₀,U₁,…,u_n)∈[0,∞)ⁿ⁺², f(t+ω,u₀, u₁,…,u_n)=f(t, u₀, u₁,…,u_n).When p(t)=1 or n=0, many scholars have studied these problems. Some sufficient criteria are established for the existence of positive periodic solution to theimpulsive delay differential equation. This chapter mainly uses the fixed point theoremof cone expansion and compression. The obtained results extends and improve mainconclusion in the corresponding references.Chapter 2 mainly considers the following nonlinear impulsive delay differentialequationWhere a∈C(R×R⁺,R⁺),Ï„_j∈C(R, R) (j=1, 2,…,n) areω-periodic function; f∈C(R×(R⁺)ⁿ,R⁺),(?)(t, u₁,…,u_n)∈[0,∞))ⁿ⁺¹,f(t+ω,u₁,…,u_n)=f(t,u₁,…,u_n).and is local Lebesgue summable function. I_j∈C(R⁺, R⁺), and there exists m＞0such that I_j+m(x)=I_j(x),t_j+m=t_j+ω.ωis a positive constant.When a(t,x(t)) is linear about x(t) or n=1 or I_j=0, many scholars haveinvestigated these problems. The sufficient conditions are obtained for the existenceof positive periodic solution to the impulsive delay differential equation. This chapteruses the fixed point theorem. The obtained results extend and improve some knownresults in the corresponding references.Some parts of this paper has been online and accepted by the Nonlinear Anal.