数学硕士论文——微分方程的研究问题《两类非线性微分方程边值问题的正解》
摘要 5-6
ABSTRACT 6
目录 7-8
第一章绪论 8-12
1.1半直线上边值问题的发展现状 8-9
1.2三阶半正边值问题的发展现状 9-12
第二章半直线上的二阶奇异边值问题 12-28
2.1介绍 12-13
2.2核函数的性质 13-14
2.3预备知识 14-24
2.4正解的存在性 24-27
2.5例子 27-28
第三章一类三阶半正边值问题的正解 28-38
3.1介绍 28-29
3.2预备知识 29-35
3.3正解的存在性 35-37
3.4例子 37-38
第四章总结 38-40
参考文献 40-42
致谢 42
【摘要】 微分方程的研究是数学研究中一个很重要的内容,同时物理、生物学、自动控制、电子技术等领域中都提出了大量微分方程问题,在解决实际问题时通常建立相应的微分方程.近一段时间以来,半直线问题已受到人们的广泛关注,如文献[1-6],但是人们对连续函数的研究相对多一些,最近也有一些文章研究关于半直线上多点边值,例如文献[7],而对于奇异性的研究相对少,这是由于半直线上奇异方程的研究比较复杂.本文第一章对半直线的现状进行了简要的说明,第二章主要讨论半直线上奇异二阶两点边值问题,主要方法是将文献[8]中的条件推广到相应线性算子的第一特征值,应用不动点指数理论证明其正解的存在性并给出了满足方程条件的具体例子.本文的第三章主要研究的是三阶半正边值问题,本文是在文献[9]研究的基础上又增加了一个非线性项,使所研究的方程更具有广泛性,利用文献[10]中类似的方法应用不动点指数理论证明了其正解的存在性,推广了文献[9]的结果,同时也给出了满足方程条件的具体的例子.
数学硕士论文代写【Abstract】 The study of differential equations is very important in mathematical problems. At the same time, the differential equations also exist in natural science such as physics, biology, automatic model in the light of actual problems, that is to say we build the differential equations.Recently, the existence of positive solutions for nonlinear second-order two-point boundary value problems on half-line has been studied by many authors [1-6]. But most of them are interested in continuous function. Now m-point problems on half-line have been studied in some article. However, the study of singularity is relative few because of the complexity of the equation.In the chapter 1 of this thesis, we explain the present the conditions of the problems. In the chapter 2, we consider the singular equation on half-line. Concerning the first eigenvalue of relevant linear operator and by means of fixed point index theory, the existence of positive solutions is obtained and the example is given.In the chapter3 of this thesis, we consider the three-order and half-positive equation at the basic of [9], we add another function in the equation so that the equation is more universality. The existence of positive solutions is obtained by means of fixed point index theory and the methods similar to those of [10]. The conclusions essentially extend and improve the main results of [9] and the example is given.
【关键词】 锥； 半直线； 半正； 正解； 不动点指数；
【Key words】 cone； half-line； half-positive； positive solutions； fixed point index；