奇异三阶积分边值问题正解的全局分歧

沈文国

(兰州工业学院基础学科部，甘肃 兰州 730050)

奇异三阶积分边值问题正解的全局分歧

沈文国

(兰州工业学院基础学科部，甘肃 兰州 730050)

研究带Riemann-Stieltjes积分边值条件的奇异三阶积分边值问题正解的全局分歧结构.首先，利用相关文献，获得了此类问题的格林函数并推证其满足的性质，同时可获得此类问题等价于一个全连续算子方程；其次，在满足所给的条件时，利用Krein-Rutmann定理建立了此类问题对应的线性问题存在简单的主特征值；最后，当非线性项在零和无穷远处满足非渐进线性增长条件、参数满足不同范围的值时，利用Dancer全局分歧定理、Zeidler全局分歧定理和序列集取极限的方法，建立了此类问题正解的全局结构，进而获得了正解的存在性.

奇异三阶积分边值问题；全局分歧；正解

1 引言

2009年，文献[1]研究了下列三阶非局部边值问题：

受上述文献的启发，本文研究下列奇异三阶积分边值问题：

正解的全局分歧结构，其中a(t)在t=0和t=1处具有奇异性，r∈(0，∞)是一个参数，

注 1.1对于用分歧技巧研究其它的正解和结点解的存在性和多解性，可参考文献[10-16].

2 格林函数的性质及推论

3 预备知识

4 主要结果

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Global bifurcation of positive solutions for singular third order problems involving Stieltjes integral conditions

Shen Wenguo
(Department of Basic Courses，Lanzhou Institute of Technology，Lanzhou 730050，China)

In this paper，we establish global bifurcation structure of positive solutions for a class of singular third-order boundary value problems.Firstly，according to the relevant literature，we obtain that the Green fuction and its property for the above problem.Meanwhile，we can obtain that the above problem is equivalent to the completely continuous operator equation.Secondly，we have that the above linear problem exists simple principal eigenvalue by the Krein-Rutman theorem.Finally，we establish the global bifurcation structure of positive solutions with non-asymptotic nonlinearity at or by Dancer and Zeidler global bifurcation theorems and the approximation of connected components.

third order singular boundary problems，global bifurcation，positive solutions

O175.8

A

1008-5513(2016)03-0221-14

10.3969/j.issn.1008-5513.2016.03.001

2015-05-27.

国家自然科学基金(11561038)；甘肃省自然科学基金(145RJZA087).

沈文国(1963-)，博士，教授，研究方向：非线性微分方程与分歧理论.

2010 MSC:34B09，34C10，34C23