非线性微分方程三点边值问题的正解

【作者】 郝新安；

【导师】 刘立山；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2007， 硕士

【摘要】 非线性泛函分析是数学中的一个重要分支，因其能很好的解释自然界中的各种各样的自然现象受到了国内外数学界和自然科学界的重视。非线性边值问题源于应用数学，物理学，控制论等各种应用学科中，是目前分析数学中研究最为活跃的领域之一。其中，多点边值问题来源于应用数学的各个领域以及物理学中的模型，具有重要的理论意义和应用价值。本文利用锥理论，不动点理论，拓扑度理论并结合上下解方法等，研究了几类非线性微分方程三点边值问题解的情况，得到了一些新成果。根据内容本文分为以下四章：在第一章中，我们综合利用schauder不动点定理，上下解方法和拓扑度理论，讨论了一类二阶三点非齐次边值问题正解的存在性，非存在性和多解性，其中b＞0，0＜η＜1，0＜αη＜1，在f具有一定单调性的条件下，我们得到了如下结果：存在b^*＞0使得当0＜b＜b^*时，边值问题（P_b）至少有两个正解，当b=b^*时边值问题（P_b）至少有一个正解，当b＞b^*时边值问题（P_b）无解。本章结果本质的改进丁文[9]的结果且证明方法也与[9]明显不同。在第二章中，我们利用Krasnosel’skii-Duo不动点定理，研究了如下n阶奇异非局部边值问题正解的存在性和多解性，其中0＜η＜1，0＜αη^n-1＜1，a（t）允许在t=0和t=1处奇异。在一定的条件下，我们得到边值问题（2．1．1）一个正解和两个正解的存在性结果。本文定理在较宽泛的条件下改进和推广了文[24]中的定理。在第三章中，通过相应线性算子的第一特征值，利用锥上的不动点指数理论，我们研究了如下n阶奇异非局部特征值问题正解的存在性，其中0＜η＜1，0＜αη^n-1＜1，a（t）允许在t=0，1处奇异且f（t，u）允许在u=0处奇异。我们得到了使特征值问题（3．1．1）正解存在的λ的具体区间。本章是对第二章的深化和继续。在第四章中，通过建立特殊的锥K₁×K₂并利用不动点指数理论，我们得到了二阶常微分方程组边值问题正解的存在性，其中α，η∈（0，1）为给定常数，f_i∈C（I×R⁺×R⁺，R⁺）（i=1，2），I=[0，1]，R⁺=[0，+∞)。其中，在所给条件中，我们使用了对应线性问题u″（t）+λu（t）=0，t∈（0，1）；u′（0）（?）0，u（1）=αu（η）的第一特征值。在方程组（4．1．1）中，非线性项f₁，f₂一个为超线性另一个为次线性，故本章结果难以在乘积空间中构造单个锥利用锥拉伸和压缩不动点定理来证明。更多还原

【Abstract】 Nonlinear functional analysis is an important branch of mathmatics, and it can explain several kinds of natural phenomena. The boundary value problems （BVPs） for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory, it is at present one of the most active fields in analyse mathematics. Among them, muliti-point BVPs come from a lot of branches of applied mathematics and physics, and it is very meaningful in both practical and theoretical aspects. The present paper employs the cone theory, fixed point theory, topological degree theory and upper-lower solutions method and so on, to investigate the existence of positove solutions to three-point BVPs of some kinds of nonlinear differential equations. The results obtained are either new or essentially generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into four chapters according to contents.In chapter 1, we consider the the existence, nonexistence and multiplicity of positive solutions for second order three-point nonhomogeneous BVPwhere b > 0, 0 <η< 1, 0 <αη< 1. Under the conditions of f is monotone, we show that there exists a positive number b* such that the problem （P_b） has at least two positive solutions for 0 < b < b*, has at least one positive solution for b = b* and no solution for b > b* by using Schauder fixed-point theorem, the upper-lower solutions method and topological degree theory. The theorem improves and generalizes the results of [9] essentially and the method is significantly different from [9].In Chapter 2, by using the Krasnosel’skii-Guo theorem on cone expansion and compression, we study the existence and multiplicity of positive solutions for a class of nth-order singular nonlocal BVPwhere 0 <η< 1, 0 <αη（n-1） < 1. The singularity may appear at t = 0,1. In Chapter 3, the nonlinear nth-order singular nonlocal BVPis considered under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, where 0 <η< 1, 0 <αη^n-1 < 1, A > 0, a may be singular at t = 0 and/or t = 1, f（t, u） may also have singularity at u = 0. The existence of positive solutions is obtained by means of fixed point index theory on cones. The explict interval of A is also given.In Chapter 4, the second-order three-point BVP systemare considered under some conditions concerning the first eigenvalue of the relevant linear problem. By constructing a special cone K₁×K₂ which is the Cartesian product of two cones and computing the fixed point index in K₁×K₂, the existence of positive solution for the system is established. It seems to be difficult to construct a single cone in product space and prove our result by using the norm-type cone expansion and compression theorem as was done in some previous papers.更多还原