Solutions For Some High-order Ordinary Differential Equations Boundary Value Problems

We mainly study the existence,uniqueness and multiplicity of solutions to highorder differential equations boundary value problems in this paper. This thesis ismainly composed of two chapters in which we discuss solutions for some high-orderordinary differential equations boundary value problems. In Chapter I, we mainly usethe fixed point theorems to discuss the existence of positive solutions for fourth-ordernonlinear differential system. We establish some sufficient conditions on nonlinearityf, g which are able to guarantee that the problem has at least a positive solution. InChapterâ…¡, we mainly use the strongly monotone operator principle and the criticalpoint theory to discuss the existence,uniqueness and multiplicity of solutions for a kindof high-order ordinary differential equation systems boundary value problems Undercertain conditions on functional F, we establish conditions such that the problem hasthe existence,uniqueness and multiplicity of solutions,respectively.In the following, we state the main results of this thesis concretely.In Chapterâ… , we mainly consider the following fourth-order nonlinear differentialequation systems boundary value problem (BVP):where f,g∈C([0, 1]×R₊⁴,R₊),R₊=[0, +∞).The main results can be stated as the following.Theorem 1.1.1 Suppose that(H₁) f₀= f_∞=g₀=g_∞=+∞,(H₂) there existρ＞0,such that f ( t, w, z, u, v)＜4ρ, g( t, w, z, u, v)＜4ρ, ( t, w, z, u, v)∈[0,1]×[0,ρ/8]×[0,ρ/8]×[0,ρ]×[0,ρ],Then BVP (1.1.1) has at least two positive solutions, i.e.,(u₁, v₁), (u₂, v₂), such that 0＜‖(u₁,v₁)‖＜ρ＜‖(u₂,v₂)‖.Theorem 1.1.2 Suppose that(H₃) f⁰=f^∞=g⁰=g^∞=0,and for all R＞0, f, g∈[0, 1]×R¹Ã—R¹Ã—[0, R]×[0,R]are bounded;(H₄) there existρ₁＞0,such that D₁ hold f(t, w, z, u, v)＞16/3ρ₁, g(t, w, z, u,v)＞16/3ρ₁, where D₁ = {(t, w, z, u, v):t∈[1/4, 3/4], w+z∈[11/1536ρ₁,ρ₁/8], u+v∈[ρ₁/4,ρ₁]}. Then BVP (1.1.1) has at least two positive solutions, i.e., (u₁, v₁), (u₂, v₂),such that 0＜‖(u₁,v₁)‖＜ρ₁＜‖(u₂,v₂)‖.In Chapterâ…¡, we mainly discuss the following boundary value problems (BVP):where F∈C¹(R², R).In these chapter, wu shall consider the existence,unique,multiplicityof solutions to the high- order boundary value problem(2.1.1)We have the following results.Theorem 2.1.1 If there exists a∈[0,Ï€^2m),such that ((?)F(u)-(?)F(u))·(u-v)≤a|u-v|²,u,v∈R²,Then BVP (2.1.1) has a unique solution in C^2m [0, 1]×C^2m[0, 1].Theorem 2.1.2 Assum F(u)≤au²+b|u|^2-γ+c, u∈R²,where a∈(0,Ï€^2m/2),γ∈(0,2),b,c＞0,then BVP (2.1.1) has at least one nonzero solution in C^2m[0, 1]×C^2m[0,1].Theorem 2.1.3 Suppose that(A₁) there existμ∈(0, 1/2) and R＞0,such that F(u)≤μ(?)F(u)·u, |u|≥R;(A₂) limsup_u→0F(u)/|u|²ï¼œÏ€^2m/2及lim inf_{|u|→∞}F(u)/|u|²ï¼žÏ€^2m/2.then BVP(2.1.1) has at least one nonzero solution in C^2m[0,1]×C^2m[0,1].Theorem 2.1.4 Suppose that(A₃) there existμ∈(0,1/2)and R＞0,such that 0＜F(u)≤μ(?)F(u)·u, |u|≥R;(A₄) lim_|u|→0F(u)/|u|²ï¼œÏ€^2m/2.then BVP (2.1.1) has at least one nonzero solution in C^2m[0, 1]×C^2m[0, 1].Theorem 2.1.5 Suppose that F is odd,i.e, F(-u) = F(u), u∈R². And supposethat the condition (A₁)in Theorem 2.1.3 is satisfied ,and that(A₅) lim sup_u→0F(n)/|u|²ï¼œÏ€^2m;(A₆)lim_{|u|→+∞}F(u)/|u|²=+∞.Then BVP (2.1.1) has infinitely many solutions in C^2m[0, 1]×C^2m[0, 1].