Existence of positive solutions for a semipositone fractional differential equation with perturbed term

Abstract

In this work, we consider the following nonlinear fractional differential equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -D^{\nu } u(t)=\lambda f(t,u(t))+e(t) \ \ \ in \ \ (0,1),\\ u^{(j)}(0)=0, \ \ 0\le j \le n-2, \ \ [D^{\alpha } u(t)]_{t=1}=0,\\ \end{array}\right. } \end{aligned}$$

where $\lambda >0$ is a parameter, $n \ge 3$, $n-1< \nu < n$, $1 \le \alpha \le n-2$ and $D^{\nu }$ stands for the standard Reimann–Liouville derivative, $f: [0,1]\times [0,+\infty ) \longrightarrow {\mathbb {R}}$ is sign-changing continuous function (that is, we have a so-called equation of semipositone problems). The perturbed term $e: (0,1) \rightarrow {\mathbb {R}}$ is measurable function and verifies some appropriate conditions. We derive some intervals of $\lambda$ such that the problem has positive solutions. Our study relies on Guo-Krasnoselskii fixed point theorem.