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Positive solutions of nonlinear boundary value problems.

机译：非线性边值问题的正解。

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摘要

In this dissertation, we study the existence, uniqueness, multiplicity, and nonexistence of positive solutions of several types of nonlinear boundary value problems (BVPs). We first investigate 2nth-order scalar periodic BVPs in the form of &parl0;aD2+b&parr0;nu=w tft,u ,t∈0,w , uk 0=uk w,k= 0,1,&ldots;,2n-1, where n ∈ N is a positive integer and D = d/dt is the derivative operator. By constructing a higher order Green's function and applying the fixed point theory on cones, we establish a series of criteria for the problem to have one, two, an arbitrary number, and even an infinite number of positive solutions.;These criteria are further extended to 2nth-order system periodic BVPs in the form of &parl0;AD2+B&parr0;nu=W tft,u ,t∈0,w , uk 0=uk w,k= 0,1,&ldots;,2n-1, where A, B, and W are m x m matrices, and f is a m-dimensional vector valued function. Moreover, by establishing the existence of a positive eigenvalue of an associated linear-system Sturm-Liouville problem using the Krein-Rutman Theorem, we obtain a sharper condition for the existence of a positive solution.;To see the behavior of the solution of periodic BVPs as the Green's function approaches singularity, we study the dependence of the unique positive solution of the periodic BVPs in the form of -u''+r2u=w tus,s∈ 0,1, u0=u1 ,u'0 =u'1, on the parameter rho as rho → 0. We show that this solution u(t; rho) has singularity of the order of rho--alpha with alpha = 2/(1 -- s) as rho → 0. Furthermore, the graph of u( t; rho) has fluctuation in t only of the order O(rho--alpha+2). The order of the solution will help us determine the location of the solution in numerical computations to study the behavior of the solution when rho is sufficiently close to 0.;The last BVP we study in this dissertation consists of the equation with p-Laplacian -&parl0;f&parl0;qt u'&parr0;&parr0;'=w tft,u ,t∈0,1 , and the BC &parl0;qu'&parr0;0 =0,u1+9 &parl0;qu'&parr0;1 =0, where &phis;(u) = |u| p--1u with p > 0. While this problem does not have a Green's function, by creating a "Green-like" functional and applying the fixed point index theory, we obtain a series of of criteria for the problem to have one, two, an arbitrary number, and even an infinite number of positive solutions. Criteria for the nonexistence of positive solutions are also derived. Parallel results for partial differential equations with p-Laplacian are given as applications of the criteria.
机译：本文研究了几种类型的非线性边值问题（BVP）的正解的存在性，唯一性，多重性和不存在性。我们首先以＆parl0; aD2 + b＆parr0; nu = w tft，u，t∈0，w，uk 0 = uk w，k = 0,1，＆ldots ,, 2n-1的形式研究2n阶标量周期BVP ，其中n∈N是一个正整数，D = d / dt是微分算子。通过构造更高阶的格林函数并将不动点理论应用在锥上，我们为该问题建立了一系列准则，该准则具有一个，两个，任意个数甚至是无限个正解。这些准则进一步得到扩展到以＆parl0; AD2 + B＆parr0; nu = W tft，u，t∈0，w，uk 0 = uk w，k = 0,1，＆ldots;，2n-1形式的2阶系统周期BVP A，B和W是mxm矩阵，f是m维向量值函数。此外，通过使用Krein-Rutman定理建立相关线性系统Sturm-Liouville问题的正特征值的存在，我们获得了存在正解的更清晰条件。随着格林函数逼近奇点而产生的BVP，我们以-u''+ r2u = w tus，s∈0,1，u0 = u1，u'0 = u的形式研究周期BVP的唯一正解的依赖性'1，在参数rho上为rho→0。我们证明该解决方案u（t; rho）具有rho-alpha阶的奇点，其中alpha = 2 /（1-s）为rho→0。 ，则u（t; rho）的图仅在t阶上具有O（rho-alpha + 2）的波动。解的阶数将有助于我们在数值计算中确定解的位置，以研究在rho足够接近0时解的行为。本文中研究的最后一个BVP由具有p-Laplacian的方程组成- ＆parl0; f＆parl0; qt u'＆parr0;＆parr0;'= w tft，u，t∈0,1和BC＆parl0; qu'＆parr0; 0 = 0，u1 + 9＆parl0; qu'＆parr0; 1 = 0，其中＆phis;（u）= | u | p--1u且p>0。虽然此问题不具有格林函数，但通过创建“类似于格林”的函数并应用不动点索引理论，我们获得了该问题具有一系列条件的一系列条件，两个，任意数量，甚至无限数量的正解。还得出了不存在正解的准则。带有p-Laplacian的偏微分方程的并行结果作为标准的应用给出。

著录项

  • 作者

    Wang, Min.;

  • 作者单位

    Northern Illinois University.;

  • 授予单位 Northern Illinois University.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2011
  • 页码 108 p.
  • 总页数 108
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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