非线性奇异微分方程解的存在唯一性
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摘要
非线性微分方程奇异边值问题是微分方程理论中的一个重要课题。大多数结果给出了方程一个或多个正解的存在性。对于解的唯一性，赵增勤在文献[34]中对奇异二阶常微分方程边值问题进行了研究，但方法较为复杂。证明解的唯一性，通常的方法是我们必须证明两个解是等价的。本文利用混合单调迭代法来证明几种奇异微分方程边值问题解的存在唯一性。
全文共分成五章：
第一章作为准备知识给出了本文要用到的相关知识内容，其中包括非线性泛函分析中的锥理论及半序方法，并建立了新的混合单调算子不动点定理。
第二章利用混合单调迭代法来证明了奇异常微分方程及时滞微分方程边值问题正解的存在唯一性，其中包括奇异高阶常微分方程解的存在唯一性，奇异p-Laplace方程解的存在唯一性，非线性奇异二阶时滞微分方程解的存在唯一性。
第三章我们将用混合单调迭代法来证明奇异椭圆微分方程边值问题解的存在唯一性。对于非线性项具有单调性假设的一类奇异椭圆边值问题，文献[87]中，作者研究了其解的存在唯一性，而对于非线性项不具有单调性假设的这类奇异椭圆微分方程，却很少有人研究其解的存在唯一性。当然，我们所讨论的方程，其非线性项要求具有特殊的形式。
第四章我们首先利用锥不动点定理讨论了非奇异二阶脉冲微分方程狄利克莱边值问题多个正解的存在性；其次我们用锥不动点定理和Leray-Schauder型非线性抉择定理，讨论了奇异二阶脉冲微分方程狄利克莱边值问题一个及多个正解的存在性。最后，在前面的工作基础上，继续将混合单调迭代法推广到脉冲微分方程中去，从而得到了奇异二阶脉冲微分方程解的存在唯一性。
The singular boundary problems of nonlinear differential equations are important subjects in the theory of differential equations. The most results are devoted on the existence of one solution or multiple solutions. On the uniqueness of the solution, Zhao zengqin[34] has studied the singular boundary problem to the second ordinary differential equations, but the method is complicated. To prove the uniqueness, the general method is that we must prove the two solutions are equivalent. In this paper, we will prove the existence and uniqueness of solutions for singular boundary value problems of differential equations by using the mixed monotone method.
The whole contents is divided into five chapters.
In chapter 1, as the beginning of this paper, we offered some relative knowledge, such as the cone theory and semi-order method in nonlinear functional analysis. Moreover, we established a new fixed point theorem of the mixed monotone iterative operator.
In chapter 2, as the second part of this thesis, by using the mixed monotone method, we gave the existence and uniqueness of singular elliptic differential equations which including singular higher order ordinary differential equations, singular p-Laplace equations, singular second delay differential equations .
In chapter 3, as one of the main parts of this thesis, by using the mixed monotone method, we discussed the existence and uniqueness of singular elliptic boundary value problems. About singular elliptic equation with the non-increasing monotonicity assumption on the nonlinearity , the author has studied the uniqueness of solutions in the reference[87]. But when the elliptic equation without the non-increasing monotonicity assumption on the nonlinearity, few author have studied the uniqueness. Of course, the equation we discussed has special nonlinearity.
In chapter 4, we first discussed the existence of multiple positive solutions of the second order nonsingular Dirichlet boundary value problem for impulsive differential equations by using the fixed point index theorem in cones . Next, we presented some new existence results for singular boundary value problems for second order impulsive differential equations by using fixed point theorem in cones and Leray-Schauder nonlinear alternative theorem. At last, we discussed the uniqueness of solutions for singular second impulsive differential equations by using mixed monotone method.
全文共分成五章：
第一章作为准备知识给出了本文要用到的相关知识内容，其中包括非线性泛函分析中的锥理论及半序方法，并建立了新的混合单调算子不动点定理。
第二章利用混合单调迭代法来证明了奇异常微分方程及时滞微分方程边值问题正解的存在唯一性，其中包括奇异高阶常微分方程解的存在唯一性，奇异p-Laplace方程解的存在唯一性，非线性奇异二阶时滞微分方程解的存在唯一性。
第三章我们将用混合单调迭代法来证明奇异椭圆微分方程边值问题解的存在唯一性。对于非线性项具有单调性假设的一类奇异椭圆边值问题，文献[87]中，作者研究了其解的存在唯一性，而对于非线性项不具有单调性假设的这类奇异椭圆微分方程，却很少有人研究其解的存在唯一性。当然，我们所讨论的方程，其非线性项要求具有特殊的形式。
第四章我们首先利用锥不动点定理讨论了非奇异二阶脉冲微分方程狄利克莱边值问题多个正解的存在性；其次我们用锥不动点定理和Leray-Schauder型非线性抉择定理，讨论了奇异二阶脉冲微分方程狄利克莱边值问题一个及多个正解的存在性。最后，在前面的工作基础上，继续将混合单调迭代法推广到脉冲微分方程中去，从而得到了奇异二阶脉冲微分方程解的存在唯一性。
The singular boundary problems of nonlinear differential equations are important subjects in the theory of differential equations. The most results are devoted on the existence of one solution or multiple solutions. On the uniqueness of the solution, Zhao zengqin[34] has studied the singular boundary problem to the second ordinary differential equations, but the method is complicated. To prove the uniqueness, the general method is that we must prove the two solutions are equivalent. In this paper, we will prove the existence and uniqueness of solutions for singular boundary value problems of differential equations by using the mixed monotone method.
The whole contents is divided into five chapters.
In chapter 1, as the beginning of this paper, we offered some relative knowledge, such as the cone theory and semi-order method in nonlinear functional analysis. Moreover, we established a new fixed point theorem of the mixed monotone iterative operator.
In chapter 2, as the second part of this thesis, by using the mixed monotone method, we gave the existence and uniqueness of singular elliptic differential equations which including singular higher order ordinary differential equations, singular p-Laplace equations, singular second delay differential equations .
In chapter 3, as one of the main parts of this thesis, by using the mixed monotone method, we discussed the existence and uniqueness of singular elliptic boundary value problems. About singular elliptic equation with the non-increasing monotonicity assumption on the nonlinearity , the author has studied the uniqueness of solutions in the reference[87]. But when the elliptic equation without the non-increasing monotonicity assumption on the nonlinearity, few author have studied the uniqueness. Of course, the equation we discussed has special nonlinearity.
In chapter 4, we first discussed the existence of multiple positive solutions of the second order nonsingular Dirichlet boundary value problem for impulsive differential equations by using the fixed point index theorem in cones . Next, we presented some new existence results for singular boundary value problems for second order impulsive differential equations by using fixed point theorem in cones and Leray-Schauder nonlinear alternative theorem. At last, we discussed the uniqueness of solutions for singular second impulsive differential equations by using mixed monotone method.
引文
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3. D.Guo and V.Lakshmikantham,"Nonlinear Problems in Abstract Cones". Academic Press. Inc. New York, 1988.
4. R.P.Agarwal, D.O'Regan,Existence Theory for Single and Multiple solutions to Singular Positone Boundary Value Problems. Journal of Differential Equations, 175(2001):393-414.
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