Research On Existence Of Periodic Solutions And Asymptotically Almost Periodic Solutions For Several Kinds Of Differential Equations

In this dissertation, mainly by using Mawhin's continuation theorem of coincidence degree, the existence of periodic solutions for two kinds of high order functional differential equations with deviating argument are investigated, and by using the equivalent definition of asymptotically almost periodic function, the existence of asymptotically almost periodic solutions for a kind of nonlinear impulsive neutral differential equations with delays are investigated. The dissertation is organized by four parts. The full text of the specific arrangements are as follows:In the first chapter, we concisely introduce the development and basic information of periodic solutions and asymptotically almost periodic solutions of studying functional differential equations.In the second chapter, we study a kind of high order functional differential equations with deviating argument by using Mawhin's continuation theorem of coincidence degree: x(2n)(t)＋f (x(t))x'(t)＋bxr(t) +g(x(t-π1(t,x(t),x'(t)))…,x(t-πm(t,x(t),x'(t))))=p(t). (2.1.1)Introduce the following conditions:(H2.1)f, p are real continuous function on R;(H2.2) letπi(t, x, y):R3→R1 are continuous functions, and letπi be 2π-periodic for all t∈R, i.e.,πi(t+2π, x, y)=πi(t, x, y), i= 1,2,…,m;(H2.3) p(t) is a 2π-periodic function and∫2π0p(t)dt= 0;(H2.4) g is a real continuous function on Rm, and there exist constants a> 0, M> 0, such thatα≤|g(Z)|≤M for any Z∈Rm;(H2.5) b≠0 and(H2.6)The main results of chapter 2 are as follows: Theorem 2.3.1 Assume that conditions (H2.1), (H2.2), (H2.3), (H2.4), (H2.5), (H2.6) hold, then the system (2.1.1) has at least a 2πperiodic solution x(t), and there exists a constant R* such that |x(t)|＝≤*, |x(j)(t)|≤R*, j = 1,2,…, n - 1.In Chapter 3, we consider the existence of periodic solutions for a kind of high order Lienard equations with deviating argument: x(n) + f(t, x(t), x(t -π0(t)))x'(t) + g(t, x(t), x(t -π1(t)),…,x(t-πm(t))) = p(t). (3.1.1) The system that we study in this chapter includes many systems which have been investigated before.Introduce the following conditions:(H3.1) f∈C(R3, R)is a continuous function, f(t + T, x, y) = f(t, x, y), (?)(x, y)∈R2;(H3.2) g∈C(Rm+2, R) is a continuous function, and g(t + T,x0,x1,…,xm) = g(t,x0,x1,…,xm), (?)(x0,x1…,xm)∈Rm+1;(H3.3)πi∈C(R, R) (i = 0,1,2,…, m) andπi(t + T) =πi(t);(H3.4) P∈C(R, R) and p(t + T)= p(t);(H3.5) sup(t,x,y)∈R3|f(t, x, y)| = A;(H3.6) There is a constant c > 0 such that |g(t,x0,x1,…,xm) + f(t,x0,xm+1)x'| >|p(t)|∞, (?)t∈R,|xi| > c {i = 0,1,…, m, m + 1);(H3.7) The function g has the decomposition g(t, x0, x1,…, xm) = K(t, x0) +∑mi=1hi(t,xi);(H3.8) |K(t,x)|≤β1 +β2|x|, whereβ1,β2 > 0;(H3.9) |hi(t, x) - hi(t, y)|≤αi|x -y|,i = 1,…,m, whereαi > 0;(H3.10) lim|x|→∞| (hi(t,x))/2|≤γi,i = 1,…, m, whereγi > 0;(H3.11) 2 1/2∑m i=1|πi(t)|∞αi +β2T + T∑m i=1γi+ A ) or (H3.12) hold.In chapter 4, we discuss the existence of asymptotically almost periodic solutions for a kind of a nonlinear impulsive neutral differential equation with time-varying delays: and let y(t)=x'(t),then the equation(4.1.1)become the following 2n-dimensional singular impul-sive delay differential equation:Introduce the following conditions:(H4.1) 0≤πij(t)≤π,00,A=(|aijuij|)n×n,B=(|bijvij|)n×n,C=(|cijwij|)n×n；(H4.4)There exist nonnegative matrices Rk such that [Ik(x)]+≤Rk[x]+,x∈Rn,k=1,2,…；(H4.5)There exists a constantσsuch that where the scalarλ> 0;(H4.6) The inequality [λK+P+Qeλπ]z*< 0 hold, where z*= (z1,…,z2n)T∈R2n∈ΩM(D),(H4.7)σk≥1 andσkz*x> Rkz*x,k= 1,2,…, where z*x= (z1,…,zn)T.The main results of chapter 4 are as follows:定理4.3.1 Assume that conditions (H4.1)-(H4.7) hold, then the zero solution of singular im-pulsive delay differential equation (4.1.2) is globally exponentially stable in PC and the exponential convergence rate is equal toλ-σ.定理4.3.2 Assume that conditions (H4.1)-(H4.7) hold, then the zero solution of singular impulsive delay differential equation (4.1.1) is globally exponentially stable in PC1 and the expo-nential convergence rate is equal toλ-σ.定理4.3.3 Assume that conditions (H4.1)-(H4.7) hold, Then the singular impulsive delay differential equation (4.1.1) has a unique asymptotically almost periodic solution.