奇異三階積分邊值問題正解的全局分歧

沈文國

(蘭州工業(yè)學院基礎(chǔ)學科部，甘肅 蘭州 730050)

奇異三階積分邊值問題正解的全局分歧

沈文國

(蘭州工業(yè)學院基礎(chǔ)學科部，甘肅 蘭州 730050)

研究帶Riemann-Stieltjes積分邊值條件的奇異三階積分邊值問題正解的全局分歧結(jié)構(gòu).首先，利用相關(guān)文獻，獲得了此類問題的格林函數(shù)并推證其滿足的性質(zhì)，同時可獲得此類問題等價于一個全連續(xù)算子方程；其次，在滿足所給的條件時，利用Krein-Rutmann定理建立了此類問題對應的線性問題存在簡單的主特征值；最后，當非線性項在零和無窮遠處滿足非漸進線性增長條件、參數(shù)滿足不同范圍的值時，利用Dancer全局分歧定理、Zeidler全局分歧定理和序列集取極限的方法，建立了此類問題正解的全局結(jié)構(gòu)，進而獲得了正解的存在性.

奇異三階積分邊值問題；全局分歧；正解

1 引言

2009年，文獻[1]研究了下列三階非局部邊值問題：

受上述文獻的啟發(fā)，本文研究下列奇異三階積分邊值問題：

正解的全局分歧結(jié)構(gòu)，其中a(t)在t=0和t=1處具有奇異性，r∈(0，∞)是一個參數(shù)，

注 1.1對于用分歧技巧研究其它的正解和結(jié)點解的存在性和多解性，可參考文獻[10-16].

2 格林函數(shù)的性質(zhì)及推論

3 預備知識

4 主要結(jié)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.

國家自然科學基金(11561038)；甘肅省自然科學基金(145RJZA087).

沈文國(1963-)，博士，教授，研究方向：非線性微分方程與分歧理論.

2010 MSC:34B09，34C10，34C23