具有变号非线性项的分数阶微分方程边值问题正解的存在性

为了进一步研究非线性项的分数阶微分方程边值问题的性质,讨论了带有变号非线性项的(n-1,1)分数阶微分方程特征值问题正解的存在性,其中分数阶导数是Riemann-Liouville型。首先利用给定边值问题的Green函数,将微分方程转化为等价的积分方程,然后在非线性项f(t,x)满足Caratheodory条件(即任意选取变量x,非线性项f(t,x)为可测函数,对(0,1)区间内几乎所有t,非线性项f(t,x)为x的连续函数)下。通过构造适当的Banach空间,运用锥拉伸与锥压缩不动点定理和Leray-Schauder非线性抉择得出边值问题正解存在的充分条件。结果表明,非线性项f(t,x)中的t可以在(0,1)区间内任何点处具有奇性,同时还改变了使边值问题的解存在的特征值λ的取值范围。研究结果为现存结论的深入研究打下了基础。

Existence of positive solutions for boundary value problems of fractional differential equations with sign-changing nonlinear term

In order to further study the properties of boundary value problems for fractional differential equations with non-linear terms,we discusses the existence of positive solutions for eigenvalue problems of (n-1,1) fractional differential equations with sign-changing nonlinear terms, where the fractional derivative is Riemann-Liouville. First we use the Green function for a given boundary value problem and convert the differential equation into an equivalent integral equation. Then, under the condition that the nonlinear term f(t, x) satisfies Caratheodory conditions (that is to say, the non-linear term f(t,x) is measurable function when the variable x was chosen arbitrarily, and the non-linear term f(t,x) is continuous function of x when the variable t was fixed). By constructing a suitable Banach space, we use the cone-stretching and compression fixed point theorem and Leray-Schauder nonlinear selection to obtain the sufficient conditions for the existence of positive solutions of boundary value problems.The results show that the nonlinear term f(t,x) can be singular at any point t. At the same time, the range of the eigenvalue which makes the solution of the boundary value problem exist was changed.The results lay a foundation for further study of the existing conclusions.