Research On Behavior Of Periodic Solutions For Several Kinds Of Forth Order Differential Equations

Nowadays, the periodic solutions of differential equations that embody the differential system's regular change have been attracted much attention. Periodic system not only widespread exist in astronomy and economics, but also in ecology, communication theories, control theories etc.. But the behavior of periodic solutions is an important branch of functional differential equation theory. Particularly, abundant results were obtained over the recent decades. However, the results on the behavior of periodic solutions of high-order functional differential equations is few.This paper discusses several types of fourth-order differential equations with delay or impulsive effects. By means of different methods, some sufficient conditions or necessary and sufficient conditions of the systems existing one or multiple positive periodic solutions, and the periodic solution that is global stability are obtained. Then some examples are explained the feasibility of each result. The full text structures are as follows:In Chapter 1, we simply introduce the development and the known results of peri-odic solutions, delay differential equations, ecology mathematics and impulsive differential equations. Then we present some problems which will be investigated in the paper.In Chapter 2, by using Fourier series theory and techniques of real analysis inequality, we mainly discuss the existence and uniqueness of periodic solutions for a class of fourth-order CDEs: prove that under certain conditions, the system exists unique 2T-periodic solution.Introduce the following conditions: The main results of chapter 2 are as follows:Theorem 2.2.1 Setδ> 0, then the equation (2.1.1) exists third order continuous dif-ferentiable T-periodic solutions if and only if the following algebraic equations have solutions on b0, bn, ln for any natural number n:Theorem 2.2.2 In equation (2.1.1), setδ> 0, then the equation (2.1.1) exists a unique third order continuous differentiable T-periodic solutions if and only ifTheorem 2.2.3 In equation (2.1.1), setδ> 0, hj= 2λjT, rk= 2λkT(λj,λk are positive integer; j=1,2,…,m; k= 1,2,…, n), Then the equation (2.1.1) exists a unique third order continuous differentiable T-periodic solutions if one of the following conditions is satisfied: is not natural; is not natural.In chapter 3, we mainly discuss a type of four species predator-prey system with delay and HollingⅢfunctional response: Firstly, we proved that the system is uniformly persistent under appropriate conditions. Then by using the Brouwer's fixed point theorem, we investigate the existence of positive periodic solutions. Finally, by constructing appropriate Lyapunov functionals, one sufficient condition is obtained for the global asymptotic stability and uniqueness of positive periodic solutions.Introduce the following conditions:(?)∈C+,(?)>(0)> 0. (3.1.2)(H3.1)b2Ld1M/c1LB1B3+d2M/c2LB1B4,b2L>d3M/c3LB2B3+d4M/c4LB2B4,b3M<(?),b4M<(?)(H3.3)Ai>0(i=1,2,3,4).The main results of chapter 3 are following:Theorem 3.2.1 Suppose that system (3.1.1) satisfies (H3.1) and (H3.2). Then system (3.1.1) is uniformly persistent.Theorem 3.2.2 If system (3.1.1) satisfies the initial condition (3.1.2) and (H3.2), then there are at least one positive T-periodic solutions on R+4.Theorem 3.3.1 Assume that conditions (H3.1)-(H3.3) hold, then system (3.1.1) exists a unique globally asymptotical stability positive periodic solution.In chapter 4, firstly, we discuss the existence of multiple positive periodic solutions for a kind of Lotka-Volterra competition patch system with delay and two harvestings in no impulsive effect: and then promote the system (4.1.3) to the following: By using the continuation theorem of coincidence theory and analytical skills, some sufficient conditions for establishing the systems have at least four positive T-periodic solutions are obtained. Introduce the following conditions:(H4.1)ai(t),bi(t),ci(t)(i=1,2,3,4),Di(t),Hi(t),βi(t)(i=1,2)are positive continuous T-periodic functions;(H4.2)ki(s)≥0 on[-τi+2,0](0<τi+2<∞),and ki(s) is a piecewise continuous and normalized function such that∫-τi+20ki(s)ds=1,i=1,2;(H4.3)a1l>c1uM2+D1u+2(?),a2l>c2uM3+D2u+2(?);(H4.4)H1l-D1uM1>0,H2l-D2uM1>0;(H4.5)a3l-c3ul1+>0,a4l-c4ul2+>0;(H4.6)There exists a positive integer p,such that tk+p=tk+T,bi(k+p)=bik,i= 1,2,3,4.(H4.7)r1l>c1uN2+D1u+2(?),r2l>c2uN3+D2u+2(?);(H4.8)H1l-D1uN1>0,H2l-D2uN1>0;(H4.9)r3l-c3uL1+>0;(H4.10)r4l-c4uL2+>0;The main results of chapter 4 are as follows:Theorem 4.2.1 Assume that conditions(H4.1)-(H4.5)hold,then system(4.1.3) has at least four positive T-periodic solutions.Theorem 4.3.2 In addition to(H4.1),(H4.2),assume further that(H4.6)-(H4.10) hold,then system(4.1.5)has at least four positive T-periodic solutions.Corollary 4.3.1 In addition to(H4.1)-(H4.4),assume further that system(4.1.3) satisfies(H',4.5)a3l/c3u>a1u/b1l,a4l/c4u>a2u/b2l.Then system(4.1.3)has at least four positive T-periodic solutions.Corollary 4.3.1 In addition to(H4.1),(H4.2),(H4.6)-(H4.8),assume further that system(4.1.5)satisfies(H'4.9)r3l/c3u>r1u/b1l,r4l/c4u>r2u/b2l.Then system(4.1.5)has at least four positive T-periodic solutions.In chapter 5,we have carried on the subtotal to the paper and presented further research problems.