【作者】 张丽娟；

【导师】 李永祥；

【作者基本信息】 西北师范大学 ， 基础数学， 2022， 硕士

【摘要】 本文应用非线性泛函分析的理论工具与方法,讨论了完全四阶常微分方程μ^（4）（x）=f（x,μ（x）,μ’（x）,μ’’（x）,μ’’’（x））,x∈R奇2π-周期解的存在性与唯一性,其中f:R⁵→R连续,f（x,y₀,y₁,y₂,y₃）关于x以2π为周期.本文的主要工作如下:1.应用Fourier分析方法,建立了对应线性方程解的L²-估计,在此基础上,应用Leray-Schauder不动点定理,在非线性项f满足适当的一次增长条件下获得了非线性方程奇2π周期解的存在性与唯一性结果.2.在非线性项f关于μ,μ’,μ’’,μ’’’可一边超线性增长且关于μ’’’满足Nagumo型增长条件的情形,应用Leray-Schauder不动点定理与先验估计技巧获得了方程奇2π-周期解的存在性与唯一性结果.3.在f关于μ,μ’,μ’’,μ’’’超线性增长且关于μ’’’满足Nagumo型增长条件情形,应用锥上的不动点指数理论,获得了方程非平凡奇2π-周期解的存在性结果.4.在f关于μ,μ’,μ’’,μ’’’次线性增长情形,应用锥上的不动点指数理论,获得了方程非平凡奇2π-周期解的存在性结果.更多还原

【Abstract】 In this thesis,some theoretical tools and methods of nonlinear functional anal-ysis are used to study the existence and uniqueness of odd 2periodic solutions of the complete fourth order ordinary differential equation μ^（4）（x）=f（x,μ（x）,μ’（x）,μ’’（x）,μ’’’（x））,x∈R, where f:R⁵→R is continuous,and f（x,y₀,y₁,y₂,y₃） is 2π-periodic in x.The main work is as follows:1.Using the Fourier analysis method,the L²-estimation of the solution of the corresponding linear equation is established.Basing upon this estimation,the ex-istence and uniqueness results of the odd 2π-periodic solutions of the nonlinear equation are obtained by using the Leray-Schauder fixed point theorem under the condition that the nonlinear term f satisfies the appropriate first-order growth con-dition.2.Under the case that the nonlinear term f can grow superlinearly on,μ,μ’,μ’’,μ’’’ at one side and it satisfies the Nagumo-type growth condition on μ’’’,the existence and uniqueness results of odd 2π-periodic solutions of the equation are obtained by using Leray-Schauder fixed point theorem and a priori estimation technique.3.In the case that f grows superlinearly on μ,μ’,μ’’,μ’’’ and it satisfies Nagumo-type growth condition on μ’’’,the existence results of nontrivial odd 2π-periodic solutions of the equation are obtained by applying the fixed point index theory in cones.4.In the case that f rows sublinearly on μ,μ’,μ’’,μ’’’ the existence results of nontrivial odd 2π-periodic solutions of the equation are obtained by applying the fixed point index theory in cones.更多还原