Existence And Multiplicity Of Solutions To High-order Differential Equation Periodic Boundary Value Problems

We mainly study the existence and multiplicity of solutions to high-order differ-ential equations in this paper. This thesis is mainly composed of two chapters in whichwe discuss some fourth-order periodic boundary value system and high-order periodicboundary value problem for nonlinear ordinary differential equations. In Chapter I, weshall construct a cone which is the Cartesian product of two cones and change the or-dinary differential problem into the fixed point problem in the cone constructed, thenuse the fixed point index theory to contain the results. In Chapterâ…¡, we study themultipliciW of nontrivial solutions for the high-order periodic boundary value prob-lem. Making use of the theory of fixed point index in cone and Leray-Schauder degree,under general conditions on nonlinearity, we prove that there exist at least six dif-ferent nintrivial solutions for the high order periodic value problem. Furthermore, ifthe nonlinearity is odd, we obtain that there exist at least eight different nontrivialsolutions. As we know, there is rare study for the two classes periodic boundary valueproblems which mentioned in the above. We discuss two classes problems and obtainsatisfied results.In Chapterâ… , we mainly consider the existence of positive solutions for the fol-lowing fourth-order periodic boundary value system:where f_i∈C([0,1]×R⁺,R⁺), h_i∈C(R⁺×R⁺,R⁺),i=1,2,R⁺=[0,+∞),α,β∈R¹and satisfyβ＞-2Ï€², 0＜α＜(β/2+2Ï€²)²,α/Ï€⁴+β/Ï€²+1＞0.Concerning the nonlinearity f_i, h_i,we assume that they satisfy the following con-ditions:(A₁)(?)f₁(t,u)/u＜α＜(?)f₁(t,u)/u;(A₂)(?)f₂(t,v)/v＞α＞(?)f₂(t,v)/v;(A₃)(?)h₁(u, v)/u=0,uniformly with respect to v∈R⁺;(A₄)(?)h₂(u, v)/v=0,uniformly with respect u∈R⁺, and (?)h₂(u, v)=0,uniformly with respect to v∈[0, N],(?)N＞0. Theorem 1.3.1. Suppose f_i∈([0, 1]×R⁺,R⁺), h_i∈C(R⁺×R⁺,R⁺), andf_i, h_i satisfy the conditions (A₁)-(A₄),i=1,2,then system (1.1.1) has at least onepositive solution.In Chapterâ…¡, we mainly consider the following high-order periodic boundaryvalue problem:where L_2mu(t)=(-1)^mu^2m(t)+∑_k=0^m-1kα_ku^2k(t) is 2m-order linear differentialoperator,f∈C(R, R).Let p(x)=(-1)^mx^2m+∑_k=0^m-1(-1)^ka_kx^2k.We assume p(2nÏ€i)≠0,n∈N={0, 1, 2...},where i the imaginary unit.Concerning the nonlinearity f,we assume that it satisfies the following conditions:(B₁) f(0)=0, f(u)u＞0, u≠0;(B₂) f is differential at 0,α₀=f′(0)＞0,and there exists a positive integer n₀such thatλ_2no-1＜α₀＜λ_2no,where {λ＿n}_n=0^∞is a sequence of all positive in {p(2nÏ€i)}_n=0^∞withλ_n-1＜λ_n for all n∈N;(B3)There existsα₁＞0,such that(?)|f(u)-α₁u|/|u|=0,and there exists a positive integer n₁ such thatλ_2n1-1＜α₁＜λ_2n1;(B₄) There exists a constant T＞0 such that|f(u)|＜a₀T, |u|≤T.The main results can be stated as the following.Theorem 2.3.1. If conditions (B₀)-(B₄) hold, then the periodic value problem(2.(?).1) have at least six different nontrivial solutions: two positive solutions, two neg-ative solutions and two sign-changing solutions.Theorem 2.3.2. If conditions (B₀)-(B₄) hold,and f is odd, i.e.,f(-u)=-f(u), u∈R,then the periodic value problem (2.(?).1) have at least eight different nontrivial solu-tions: two positive solutions, two negative solutions and four sign-changing solutions.