数学硕士论文代写节选—《不动点定理和奇异方程的非平凡解》
目录
摘要 5-7
Abstract 7-8
第1章绪论 10-16
第2章半闭1-集压缩算子的不动点定理 16-30
2.1引言 16
2.2预备知识 16-19
2.3几个新的不动点定理 19-30
第3章一类奇异方程的非平凡解 30-40
3.1引言 30-31
3.2预备知识 31-33
3.3存在定理 33-40
第4章总结 40-42
参考文献 42-44
致谢 44
【摘要】 半闭1-集压缩算子是一类新兴的算子,许多人对它感兴趣,在这里我们主要讨论的是关于半闭1-集压缩算子的不动点定理的问题.本文第2章研究的是在Banach空间的半闭1-集压缩算子的不动点定理.关于这类算子的研究方法有很多,本文则与以往论文略有不同.以往的这类文献中所提的条件及关系式都是以范数形式给出的,而这里通过定义在锥上的某一泛函ρ来代替以往常用的范数.我们利用这个泛函给出的不等式通过不动点指数理论,最终获得了一些新的不动点定理.同时,本文在许绍元的论文[4]的基础上,作了补充.除了讨论在α≥1,β>0的条件下,不动点的存在情况；还讨论了在0<α<1,β≤0或α<0的条件下,不动点是否存在.多年来,常微分方程及方程组的非线性边值问题一直是人们研究讨论的焦点,在非线性微分方程边值问题的解的存在性研究中赋予边值各种不同的条件,得到许多结果.人们尤其对奇异非线性二阶两点边值问题的正解存在性感兴趣,而本文第3章主要研究的是奇异次线性二阶多点边值问题的非平凡解.在以往的这类文章中都有个条件,要求u≥0时,f(u)≥0.而在本文中这个条件发生改变,f(u)可以为负,也就是说本文是一个奇异半正次线性二阶多点边值问题.奇异次线性二阶m-点边值问题：其中在x=0和x=1是允许奇异的,f不必非负.以往解这类问题多通过上下解,Schauder不动点定理或不动点指数的方法,本文则是通过在一些条件下关于相关线性算子的第一特征值的拓扑度理论来获得非平凡解的存在性.
数学硕士论文代写【Abstract】 Semiclosed 1-set-contractive operators are a class of new operators, and many people are interested in these operators. We mainly discuss fixed point theorems of semiclosed 1-set-contractive operators.In the second chapter of this thesis, we study fixed point theorems of semiclosed 1-set-contractive operators in Banach spaces. There are many research methods about this class of operators, but the method in this thesis is a little different from others. The conditions and relevant expressions in the literature in the past are in the form of norm, but here we replace norm in common use with some convex functional p defined on a cone. We use inequalities given by this functional to obtain some new fixed point theorems by. fixed point index theory.At the same time, the third chapter of this thesis makes some complements of the results in Shaoyuan Xu [4]. Besides discussing the existence of fixed point under the condition ofα≥1,β>0, we also discuss whether the fixed point exists under the conditions of 0<α<1,β≤0 orα<0.Recent years, nonlinear boundary value problems of singular ordinary differential equations and systems are always the focal points that are studied and discussed. People endow the boundary with various different conditions when considering the existence of solutions, then obtain many results. People are interested in the existence results of positive solutions for singular second-order second-point boundary value problems. In the third chapter of this thesis, we are concerned with nontrivial solutions for singular sublinear second-order multi-point boundary value problems.It is necessary when u≥0, f(u)≥0 // in this class of papers in the past. But this condition is weakened in this thesis, that is,f can be nonnegative.Singular sublinear second-order m-point boundary value problems: where h(x) is allowed to be singular at x=0 and x=1, and f is not necessary to be nonnega-tive.In the preceding works, this problem is studied by the methods of upper and lower solutions, Schauder’s fixed point theorem or the fixed point index, but we obtain the existence results. of nontrivial solution, by means of the topological degree theory under some conditions concerning the first eigenvalue corresponding to the relevant linear operator.
【关键词】 半闭1-集压缩； 不动点； 奇异； 次线性； 非平凡解；
【Key words】 semiclosed 1-set-contractive； fixed point； singular； sublinear； nontrivial solutions；