Research On Existence Of Anti-periodic Solutions For Several Kinds Of Differential Equations

In this dissertation, the existence of anti-periodic solutions for several types of differential equations with delay or impulsive effects are investigated. By means of different research methods, several kinds of the sufficient conditions of the systems existing anti-periodic solutions have been obtained. The full text of the specific arrangements are as follows:In Chapter 1, we simply introduce the development and basic information of periodic and anti-periodic solutions.In Chapter 2, we use the Leray-Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a kind of Lienard equation with infinite delays of the form:X"(t)＋f1(t,x(t))x'(t)＋f2(x(t))(x'(t))2 Some conclusions extend the results of [22,33].Introduce the following conditions:There exist constants A,L1,L2,H,B,F1,F2>0, for all t∈[0,T], x,u,v∈R, such that:The main results of chapter 2 are as follows:Theorem 2.3.1 Assume that (H2.1),(H2.2),(H2.3),(H2.5),(H2.6) hold, then the system (2.1.2) has a unique T-anti-periodic solution.In Chapter 3, we discuss the sufficient conditions for the existence and exponential stability of anti-periodic solutions for cellular neural networks (CNNs) with impulses and delays as follows: prove that under certain conditions, the system (3.1.1) exists one exactly T-anti-periodic solutions. In this chapter, we promote the methods and conclusions [24,42,43,44] to the situation with delay and impulse.Introduce the following conditions:(H3.1) There exist positive constants Fj,Lj, such that:fj(0)= 0,|fj(u)|≤Fj,|fj(u)-fj(v)|≤Lj|u-v|.(H3.2) dik is a real sequence and dik> 0, i= 1,2,…, n, k= 1,2,….(H3.3) (?)00,η>0 andλ> 0, i= 1,2,…, n, and setsuch thatThe main results of chapter 3 are as follows:Theorem 3.3.1 Suppose that conditions (H3.1)-(H3.5) are satisfied. Then systems (3.1.1) has exactly one T-anti-periodic solution z*(t)={zi*(t)}. More over, zi*(t)={zi*(t)} is globally exponentially stable.In Chapter 4, we study a kind of high dimensional neutral functional differential equations with delay by using of exponential dichotomy and fixed point theorem: the sufficient conditions to ensure the existence of T-anti-periodic solutions of system (4.1.1) have been obtained.Introduce the following conditions: (H4.1) There exist positive differentiable functions d1(t), d2(t),…, dn(t)(C1≤di(t)≤C2,C1,C2 are two positive constants), continuous T-periodic function a(t), such that:(H4.2) There exist positive differentiable functions are two positive constants), continuous T-periodic function (?)(t),such that:(H4.3) q1=∫-∞0|G(s)|ds<1,q2=∫-∞0|Q(s)|ds is bounded.(H4.4) There exist positive constants L1,L2,L3, such that:L1= sup0≤t≤T|A(t)|,|g(u)-g(v)|≤L2|u-v|,|f(t,u)-f(t,v)|≤L3|u-v|.(H4.5) Assume that where k1= exp(∫0Tα(λ)dλ)<1, M=(H4.6) Assume that whereThe main results of chapter 4 are as follows:Theorem 4.3.1 Assume that conditions (H4.1), (H4.3), (H4.4), (H4.5) hold, then system (4.1.1) has one T-anti-periodic solution.Theorem 4.3.2 Assume that conditions (H4.2), (H4.3), (H4.4), (H4.6) hold, then system (4.1.1) has one T-anti-periodic solution.