几类二阶常微分方程周期边值问题正解的全局结构
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摘要
本学位论文主要运用分歧理论和线性二阶常微分方程周期边值问题谱理论研究了几类含参非线性二阶常微分方程周期边值问题正解的全局结构．全文共分为六章．
第一章是绪论．阐明本文的研究背景,介绍所研究的主要问题、所得的主要结果以及所需要的一些预备知识．
第二章运用Rabinowitz全局分歧定理[48,50]研究二阶常微分方程周期边值问题正解的全局结构以及正解分支的稳定性．这里λ∈(?)是参数．本章在非线性项f在原点处满足渐近线性增长条件下,分f在(0,+∞)有无零点及加权项a是否变号,分别获得了问题(0.0.1)正解的全局结构,并在此基础上,获得问题(0.0.1)在λ的某个区间上正解和多个正解的存在性结果．更进一步,通过给非线性项f的二阶导数附加符号条件,我们还讨论了正解连通分支的分歧方向和正解的稳定性．本章主要结果改进和推广了已有的关于二阶常微分方程周期边值问题正解存在性的诸多结果．
第三章运用分歧理论及逼近的方法,在非线性项f在原点处满足次线性增长的条件下,分f在(0,+∞)有无零点及加权项a是否变号,继续讨论问题(0.0.1)正解的全局结构．对这种情形,直接应用Rabinowitz全局分歧定理来讨论问题(0.0.1)从平凡解线上产生的正解连通分支已不可行,我们利用逼近的思想发展了一种新的方法克服了这一困难,仍然获得了问题(0.0.1)正解的全局结构．本章的结果改进和推广了文[22](J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008)中的一些主要结果,而且本章研究问题的方法也有较大创新．
第四章继续运用分歧理论及逼近的方法,在非线性项f在原点处满足超线性增长的条件下,分f在(0,+∞)有无零点及加权项a是否变号,讨论问题(0.0.1)正解的全局结构．与第三章比较,本章证明难度更大．本章的结果也改进和推广了文[22](J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008)中的一些主要结果．
第五章运用区间分歧定理[41,42]和拓扑度理论,在不存在及a(·)非负时获得了问题(0.0.1)正解的全局结构,并将该结果推广到了一类更广泛的二阶常微分方程周期边值问题与前三章不同,在本章中正解的连通分支都是从一个区间上分歧出的,而且本章研究问题的方法也是全新的．
在文献[74]中,B. P. Rynne讨论了Dirichlet边值条件下一类非线性椭圆特征值问题的无穷多个正解的存在性．受这篇文献启发,在第六章中,我们讨论了二阶常微分方程周期边值问题正解连通分支的振荡性及无穷多个正解的存在性．在问题(0.0.3)中,通过给非线性项g附加振荡性条件(例如g是周期的),获得的正解连通分支也相应地开始振荡,而且振荡的振幅有界且不趋于零．本章将文献[74]中的一些结论推广到了周期边值情形下,同时也改进和推广了文献[24]中的主要结果．
This thesis focuses on the global structure of positive solutions of some periodic boundary value problems for second order ordinary differential equations with a pa-rameter, via bifurcation techniques and the theory of spectrum of periodic boundary value problems of linear second order ordinary differential equations. It is divided into six chapters.
In chapter one, The background material and some preliminaries are introduced. The main problems as well as the corresponding results are also presented.
By using Rabinowitz global bifurcation theorems [48,50], chapter two is con-cerned with the global structure and the stability of positive solutions for periodic boundary value problems of second order ordinary differential equation whereλ∈R is a parameter. Different global structures of positive solutions for (0.0.1) are achieved with the condition that f satisfies asymptotically linear growth at the origin, according to whether f has zeros in (0,∞) and whether a is sign changing. Based on these results, the existence and multiplicity of positive solutions for (0.0.1) are obtained in an interval of A. Furthmore, by adding sign condition for the second derivative of f, we also discuss bifurcation direction and stability of the branch of positive solutions. The main results in this chapter improve and generalize many previous theroems on the existence of positive solutions.
Chapter three continue to study the global structures of positive solutions for (0.0.1). In this chapter,f is required to satisfy sublinear growth at the origin, in the case that whether f has zeros in (0,∞) or not and whether a is sign changing or not. Such circumstances, Rabinowitz global bifurcation theorem can not applied directly to discuss the branch of positive solutions for (0.0.1) bifurcated from the trivial solution. To overcome this difficulty, a brand new approach by approximation is developed, and then the global structures of positive solutions for (0.0.1) can be also achieved. The results in this chapter improve and generalize some main results of [22] (J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008).
In chapter four,∫is needed to satisfy suplinear growth condition at the origin. To achieve different global structures of positive solutions of (0.0.1) according to whether f has zeros in (0,∞) and whether a is sign changing, more difficulties is appearing compared to chapter three. The results in this chapter also improve and generalize some main results of [22](J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008).
The fifth chapter is devoted to obtaining global structures of positive solutions for (0.0.1) under the conditions that the limits lims→0+(f(s))/s and limss→∞+(f(s))/sdon't exist and the weighted term a is nonnegative. Furthermore, the results are extended to a more generalized problem The main methods applied in the chapter include topological degree theory and the bifurcation theorems from an interval[41,42], which are brand new in the sence for periodic boundary value problems. Different from the first three chapter, the branch of positive solutions bifurcates from an interval in this chapter.
Inspired by [74], in which B. P. Rynne studied the existence of infinitely many positive solutions for a semilinear elliptic problems, in the sixth chapter we pay our attention to the oscillation property of the branch of positive solutions and the existence of infinitely many positive solutions for the problem For the above problem, we show that with certain oscillation condition on the non-linearity g, the continuum of positive solutions of the problem (0.0.3) also begin to oscillate correspondingly,and the amplitude of these oscillations is bounded away from zero.The results in this chapter generalize some results of[74]to the periodic boundary value problems and improve the main conclusions of[24].
第一章是绪论．阐明本文的研究背景,介绍所研究的主要问题、所得的主要结果以及所需要的一些预备知识．
第二章运用Rabinowitz全局分歧定理[48,50]研究二阶常微分方程周期边值问题正解的全局结构以及正解分支的稳定性．这里λ∈(?)是参数．本章在非线性项f在原点处满足渐近线性增长条件下,分f在(0,+∞)有无零点及加权项a是否变号,分别获得了问题(0.0.1)正解的全局结构,并在此基础上,获得问题(0.0.1)在λ的某个区间上正解和多个正解的存在性结果．更进一步,通过给非线性项f的二阶导数附加符号条件,我们还讨论了正解连通分支的分歧方向和正解的稳定性．本章主要结果改进和推广了已有的关于二阶常微分方程周期边值问题正解存在性的诸多结果．
第三章运用分歧理论及逼近的方法,在非线性项f在原点处满足次线性增长的条件下,分f在(0,+∞)有无零点及加权项a是否变号,继续讨论问题(0.0.1)正解的全局结构．对这种情形,直接应用Rabinowitz全局分歧定理来讨论问题(0.0.1)从平凡解线上产生的正解连通分支已不可行,我们利用逼近的思想发展了一种新的方法克服了这一困难,仍然获得了问题(0.0.1)正解的全局结构．本章的结果改进和推广了文[22](J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008)中的一些主要结果,而且本章研究问题的方法也有较大创新．
第四章继续运用分歧理论及逼近的方法,在非线性项f在原点处满足超线性增长的条件下,分f在(0,+∞)有无零点及加权项a是否变号,讨论问题(0.0.1)正解的全局结构．与第三章比较,本章证明难度更大．本章的结果也改进和推广了文[22](J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008)中的一些主要结果．
第五章运用区间分歧定理[41,42]和拓扑度理论,在不存在及a(·)非负时获得了问题(0.0.1)正解的全局结构,并将该结果推广到了一类更广泛的二阶常微分方程周期边值问题与前三章不同,在本章中正解的连通分支都是从一个区间上分歧出的,而且本章研究问题的方法也是全新的．
在文献[74]中,B. P. Rynne讨论了Dirichlet边值条件下一类非线性椭圆特征值问题的无穷多个正解的存在性．受这篇文献启发,在第六章中,我们讨论了二阶常微分方程周期边值问题正解连通分支的振荡性及无穷多个正解的存在性．在问题(0.0.3)中,通过给非线性项g附加振荡性条件(例如g是周期的),获得的正解连通分支也相应地开始振荡,而且振荡的振幅有界且不趋于零．本章将文献[74]中的一些结论推广到了周期边值情形下,同时也改进和推广了文献[24]中的主要结果．
This thesis focuses on the global structure of positive solutions of some periodic boundary value problems for second order ordinary differential equations with a pa-rameter, via bifurcation techniques and the theory of spectrum of periodic boundary value problems of linear second order ordinary differential equations. It is divided into six chapters.
In chapter one, The background material and some preliminaries are introduced. The main problems as well as the corresponding results are also presented.
By using Rabinowitz global bifurcation theorems [48,50], chapter two is con-cerned with the global structure and the stability of positive solutions for periodic boundary value problems of second order ordinary differential equation whereλ∈R is a parameter. Different global structures of positive solutions for (0.0.1) are achieved with the condition that f satisfies asymptotically linear growth at the origin, according to whether f has zeros in (0,∞) and whether a is sign changing. Based on these results, the existence and multiplicity of positive solutions for (0.0.1) are obtained in an interval of A. Furthmore, by adding sign condition for the second derivative of f, we also discuss bifurcation direction and stability of the branch of positive solutions. The main results in this chapter improve and generalize many previous theroems on the existence of positive solutions.
Chapter three continue to study the global structures of positive solutions for (0.0.1). In this chapter,f is required to satisfy sublinear growth at the origin, in the case that whether f has zeros in (0,∞) or not and whether a is sign changing or not. Such circumstances, Rabinowitz global bifurcation theorem can not applied directly to discuss the branch of positive solutions for (0.0.1) bifurcated from the trivial solution. To overcome this difficulty, a brand new approach by approximation is developed, and then the global structures of positive solutions for (0.0.1) can be also achieved. The results in this chapter improve and generalize some main results of [22] (J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008).
In chapter four,∫is needed to satisfy suplinear growth condition at the origin. To achieve different global structures of positive solutions of (0.0.1) according to whether f has zeros in (0,∞) and whether a is sign changing, more difficulties is appearing compared to chapter three. The results in this chapter also improve and generalize some main results of [22](J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008).
The fifth chapter is devoted to obtaining global structures of positive solutions for (0.0.1) under the conditions that the limits lims→0+(f(s))/s and limss→∞+(f(s))/sdon't exist and the weighted term a is nonnegative. Furthermore, the results are extended to a more generalized problem The main methods applied in the chapter include topological degree theory and the bifurcation theorems from an interval[41,42], which are brand new in the sence for periodic boundary value problems. Different from the first three chapter, the branch of positive solutions bifurcates from an interval in this chapter.
Inspired by [74], in which B. P. Rynne studied the existence of infinitely many positive solutions for a semilinear elliptic problems, in the sixth chapter we pay our attention to the oscillation property of the branch of positive solutions and the existence of infinitely many positive solutions for the problem For the above problem, we show that with certain oscillation condition on the non-linearity g, the continuum of positive solutions of the problem (0.0.3) also begin to oscillate correspondingly,and the amplitude of these oscillations is bounded away from zero.The results in this chapter generalize some results of[74]to the periodic boundary value problems and improve the main conclusions of[24].
引文
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