带有变号格林函数的二阶边值问题正解

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

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.

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.