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

Abstract

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.

Keywords

• ordinary differential equation
• fixed point theorem
• Banach space
• Green's function
• positive solution
• fractional differential equations