带积分边界条件的三阶微分方程正解的存在性

Existence of positive solutions of third order differential equations with integral boundary conditions

研究带积分边界条件的三阶微分方程正解的存在性问题,通过特征值理论、α-凹算子理论和锥上的不动点定理,得到了带积分边界条件的三阶微分方程解的存在性和不存在的结果。首先,给出了格林函数相关的表达式及其相关性质;其次,构造了一个合适的锥和算子;最后,利用次线性和超线性的性质、不动点定理、特征值理论得到了正解存在和不存在时的情况。同时,将p分为3种不同情况,分别是p>1,p=1,p=∞,在p取值不同的情况下,给出了解的存在性、不存在性的结果,还得出了参数的取值范围和正解对于参数的依赖性的结果。

In this paper,the existence and nonexistence of positive solutions of third-order differential equations with integral boundary conditions are studied.By using eigenvalue theory,α-concave operator theory and fixed point theorem on cones,the existence and nonexistence of solutions of third-order differential equations with integral boundary conditions are obtained.First,we get the expression of Green's function and the properties of Green-s function.Secondly,we construct an appropriate cone and operator by calculation.Finally,the existence and nonexistence of positive solutions are obtained by using the properties of sublinear and superlinear,fixed point theorem and eigenvalue theory.At the same time,we need to be divided into three different cases in the paper,p>1,p=1,p=∞.In the case of different values,we give the results of existence and non-existence of solutions,and also obtain the value range of parameters and the results of the dependence of positive solutions on parameters.