【作者】 张祖峰；

【导师】 刘斌；

【作者基本信息】 华中科技大学 ， 应用数学， 2007， 硕士

【摘要】 常微分方程起源于应用学科,诸如核物理学,气体动力学,流体力学,非线性光学等.常微分方程边值问题是常微分方程理论研究中最重要的课题之一.常微分方程边值问题在共振条件下解的存在性近十年来受到学者的关注.但是对于高阶复杂边值问题与含p-Laplacian算子型方程边值问题在共振情况的可解性的研究还不多见.针对这种情况本文作了如下研究:第一章我们着重介绍了问题的起源和相关背景,以及问题的发展现状和趋势.第二章,我们用Mawhin重合度定理研究了共振条件下一类四阶四点边值问题的可解性,在增长条件下得到解存在的充分条件.第三章,我们讨论了一类含p-Laplacian算子型多点边值问题在共振条件下的可解性,用Leray-Schauder度与Brouwer度得出解存在的两个充分条件,我们的结果推广改进了现有文献的一些结果.第四章,我们还用Leray-Schauder度与Brouwer度讨论了一类含p-Laplacian算子型多点边值问题具有两个临界条件的可解性,这个讨论方法与已有的结果的讨论方法不同.今后,我们还可以结合其他的一些工具,讨论共振问题的正解与多解的存在性,得到更全面的结果.更多还原

【Abstract】 The ordinary differential equation boundary value problem is one of the most important branches of ordinary differential equations. It is resulted from physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. Boundary value problem at resonance have been studied by several authors in recent ten years. But the existence of high-order with complex boundary value conditons and equation including p-Laplacian operator has rarely been studied. So in this paper, we consider the problem as following: In section I, we give an introduction of the origin of the problems, the status of the research on related issues, and the trend of the future development. In section 2, result for solvability of 4-point boundary value problems of fourth order differential problem at resonance is obtained by using coincidence degree theory due to Mawhin. An sufficient condition is given under non-linear growth restriction on f. In section 3, we study the existence of p-Laplace-like equtions subject to multi-point boundary value problems at resonance. By using degree theory, we obtained two sufficient conditions for solvability, some known results are improved. In section 4, we use Brouwer degree and Leray-Schauder degree to discuss solvability of multi-point boundary value problem with two critical conditions. the methods used in the paper are different from some known results.Later, we can consider boundary value problem at resonance with different tools to obtain the existence of postive solutions and multi-solutions.更多还原