东北大学学报（自然科学版）

东北大学学报:自然科学版 ›› 2020, Vol. 41 ›› Issue (4): 604-608.DOI: 10.12068/j.issn.1005-3026.2020.04.026

• 数学 • 上一篇    

带有变号格林函数的二阶边值问题正解

张国伟， 曲雪冰   

  1. (东北大学 理学院， 辽宁 沈阳110819)
  • 收稿日期:2019-09-26 修回日期:2019-09-26 出版日期:2020-04-15 发布日期:2020-04-17
  • 通讯作者: 张国伟
  • 作者简介:张国伟(1965-)，男，辽宁沈阳人，东北大学教授.
  • 基金资助:
    国家自然科学基金资助项目(61473065).

Positive Solutions of Second-Order Boundary Value Problems with Sign-Changing Green’s Function

ZHANG Guo-wei， QU Xue-bing   

  1. School of Sciences， Northeastern University， Shenyang 110819， China.
  • Received:2019-09-26 Revised:2019-09-26 Online:2020-04-15 Published:2020-04-17
  • Contact: QU Xue-bing
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摘要： 研究了一类带有变号格林函数的二阶边值问题正解的存在性，格林函数变号由边值条件中系数的不同取值所致，这与文献中通常由未知函数一次项系数的变化导致格林函数变号不同.没有非线性项非负的限制时，通过对格林函数的正部和负部赋予约束条件，证明了二阶边值问题正解的存在性.利用两个具体例子说明了理论结果的有效性，例子中边值条件的系数包含了正的和负的两种情形.另外对两类不同的边值条件给出了说明.

关键词: 正解, 变号格林函数, 二阶边值问题, 全连续算子, Leray-Schauder不动点定理

Abstract: The existence of positive solutions for a class of second-order boundary value problems with a sign-changing Green’s function was studied， and the sign-changing Green’s function was caused by different values of coefficients in boundary value conditions， which is different from that the change of the coefficient of the first order of the unknown function usually leads to the change of the Green’s function. When there is no non-negative limitation of nonlinear term， the existence of positive solutions for second-order boundary value problems was proved by giving constraints to the positive and negative parts of Green’s function. The validity of the theoretical results was illustrated by two concrete examples， in which the coefficients of boundary value condition include both positive and negative cases. In addition， two different boundary conditions were explained.

Key words: positive solution, sign-changing Green’s function, second-order boundary value problem, completely continuous operator, Leray-Schauder fixed point theorem

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