【作者】 沈建和；

【导师】 周哲彦； 余赞平；

【作者基本信息】 福建师范大学 ， 应用数学， 2005， 硕士

【摘要】 本文运用微分不等式的技巧(或称为上下解方法),一定条件下证明几类非线性边值问题(不带小参数)解的存在性(部分内容证到唯一性),同时运用上面部分存在性(或唯一性)结果处理数学物理中广泛出现带小参数的几类奇摄动边值问题,得到奇摄动边值问题解的存在性以及解的渐近估计.其内容安排如下: 第一部分,给出上、下解的概念及其“Nagumo条件”,同时给出两个二阶微分不等式的基本结果(引用文献[1][2][12]),该基本结果第二节会用到,同时也为下面内容的自然推广作铺垫. 第二部分,利用边界层函数法,构造两类带有高阶转向点的二阶非线性二次奇摄动边值问题的高阶渐近解(二次奇摄动边值问题的提法来自F.A.Howes专著[12],同时运用第一部分中引理1.4和引理1.5,得到了摄动问题解的存在性及渐近解关于精确解的误差估计. 第三部分,首先给出三阶非线性微分方程Robin边值问题解的存在唯一性,接着运用上述存在唯一性结果,处理带有转向点的三阶非线性Robin边值问题,得到其解的存在唯一性以及解的渐近估计. 第四部分,利用上、下解的方法,在一定条件下得到一类三阶非线性微分方程的非线性三点边值问题解的存在性,作为推论,讨论了一类带周期边界条件的边值问题的解的存在性. 第五部分,利用上、下解的方法,一定条件下得到一类二阶非线性微分方程组的三点边值问题解的存在性,同时将其结果作用于四阶微分方程的三点边值问题,推广了前人的一些结果.作为该结果的应用,考察了一类四阶半线性奇摄动三点边值问题,得到其解的存在性及其解的渐近估计.更多还原

【Abstract】 In this paper, existence(or uniqueness) results of some classes of BVP(without small parametre) were achieved via the method of upper and lower solution firstly, furthermore, we apply some of the existence results(or uniquness) to singlar perturbed BVP which contain small parametre, these kinds of BVP arrising frequently in mathematics and physics. By this way, existence of solution for singlar perturbed BVP can be gained, and the asymptotic behavior of solution as ε→ 0 can be studied additionly.This paper is arranged followingIn section 1, the concept of lower solution , upper solution and Nagumo condition were showed, and some princial theorems (Lemma 1.4,1.5) of second-order differential inequality were introducted, this concept and lemmas cite from ([l][2][12]),the Lemmas will be utilized in the section 2.In section 2, we construct the higher-order asymptotic solution of two kinds of nonlinear quadratic singular perturbed BVP by the method of boundary layer function,then existence of solution of nonlinear quadratic singlar perturbed BVP and the estimate of asymptotic solution about exact solution were achieved directly via the application of Lemma 1.5 which was showed in section 1.In section 3, Firstly, existence and uniqueness of solution of third order nonlinear Robin BVP was showed, then existence and uniqueness of solution of third order nonlinear Robin BVP with a turning point was achieved via the application of the existence and uniquness results mentioned above.In section 4, existence of solution of a class of nonlinear three-point BVP for nonlinear third-order differential equation was given, the analysis was based on the method of upper and lower solution. As a corollary, existence of solution of a class of third-order BVP with periodic boundary condition was gained.In section 5 , existence of solution of a class of nonlinear second-order differential systems for three-point BVP was given, the analysis was based on the method of upper and lower solution. As a corollary, existence of solution of a class of three-point BVP for forth-order differential equation was show; As an application, considered a type of forth-order semilinear singular perturbed three point BVP.更多还原