带脉冲条件的不定S-L问题的非实特征值

发布时间：2018-01-02 19:26

本文关键词：带脉冲条件的不定S-L问题的非实特征值　出处：《山东大学》2016年硕士论文　论文类型：学位论文

【摘要】：脉冲现象作为一种瞬时突变现象,在现代科技各领域的实际问题中是普遍存在的。自90年代以来,脉冲微分系统兼具连续系统和离散系统的特征。它最突出的特点是充分考虑瞬时突发现象对系统状态的影响,能够更深刻、更精确的反应事物的变化规律。脉冲条件又称为转移性条件,非连续条件,临界条件等。在数学理论上,脉冲微分系统作为一个非连续系统的分支,是对数学中连续问题的重要补充,并且反过来有利于理解连续问题。目前为止,对于带脉冲条件的Sturm-Liouville问题的研究主要是针对右定问题的研究,也就是权函数不改变符号。而本文主要研究带脉冲条件的不定Sturm-Liouville问题。本文主要研究的问题如下：第二章主要考虑的是带脉冲条件的右定Sturm-Liouville问题。同连续右定Sturm-Liouville问题的研究方法相类似,建立新的希尔伯特空间,并在其上建立与问题相对应的自伴算子,但是这里对脉冲条件中的系数作了限制,并且举例说明这一限制是必要的。其次,将连续与带脉冲条件的右定Sturm-Liouville问题做了简单比较,得到两个问题有相同特征值的充要条件并佐以例子进行了说明。第三章主要考虑的是带脉冲条件的不定Sturm-Liouville问题的非实特征值。将[34]中非实特征值存在的相关理论、性质推广到带脉冲条件的不定Sturm-Liouville问题,得到非实特征值存在的必要条件,这一结果与连续的不定Sturm-Liouville问题相类似。由于已经建立了带脉冲条件的右定Sturm-Liouville问题的特征值理论、振动理论、最小最大值理论等,因此可以利用双谱参数法,也即特征曲线法来研究非实特征值,得到带脉冲条件的不定Sturm-Liouville问题的关于非实特征值的相关结论。并在双谱参数法的基础上加以分析,得到恰好存在两个非实特征值的充分条件。第四章主要是对系数加以限制,得到对带脉冲条件的不定Sturm-Liouville问题的非实特征值界的估计。
[Abstract]:Pulse phenomenon, as a kind of transient sudden change phenomenon, is common in various fields of modern science and technology. Since 90s. Impulsive differential system has the characteristics of both continuous system and discrete system. Its most outstanding feature is that the influence of instantaneous burst phenomenon on the system state can be considered fully, so it can be more profound. Pulse condition is also called transfer condition, discontinuous condition, critical condition, etc. In mathematical theory, impulsive differential system is regarded as a branch of discontinuous system. Is an important complement to continuous problems in mathematics and, in turn, facilitates the understanding of continuous problems. The study of Sturm-Liouville problem with impulsive condition is mainly focused on the right definite problem. That is, the weight function does not change the symbol. In this paper, we mainly study the indefinite Sturm-Liouville problem with impulsive conditions. The main problems of this paper are as follows:. In chapter 2, we mainly consider the right-definite Sturm-Liouville problem with impulsive conditions, which is similar to the method of continuous right-definite Sturm-Liouville problem. A new Hilbert space is established and a self-adjoint operator corresponding to the problem is established on it. However, the coefficients in the impulsive condition are restricted here, and an example is given to illustrate that this restriction is necessary. Secondly. A simple comparison is made between the continuous Sturm-Liouville problem and the right-definite Sturm-Liouville problem with impulsive conditions. The necessary and sufficient conditions for two problems to have the same eigenvalues are obtained and illustrated by an example. In chapter 3, the nonreal eigenvalues of the indeterminate Sturm-Liouville problem with impulsive conditions are considered. [34 the theory of the existence of nonreal eigenvalues is extended to the indefinite Sturm-Liouville problem with impulsive conditions, and the necessary conditions for the existence of nonreal eigenvalues are obtained. This result is similar to that of the continuous indefinite Sturm-Liouville problem, since the eigenvalue theory of the right-definite Sturm-Liouville problem with impulsive conditions has been established. Vibration theory, minimum maximum theory and so on, so we can use bispectral parameter method, that is, characteristic curve method to study non-real eigenvalues. Some conclusions on the nonreal eigenvalues of the indeterminate Sturm-Liouville problem with impulsive conditions are obtained and analyzed on the basis of bispectral parameter method. The sufficient conditions for the existence of two nonreal eigenvalues are obtained. Chapter 4th mainly limits the coefficients. The nonreal eigenvalue bounds of the indeterminate Sturm-Liouville problem with impulsive conditions are estimated.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

【相似文献】