Existence and uniqueness of nontrivial solutions for eigenvalue boundary value problem of nonlinear fractional differential equation

Abstract

In this paper, we study the existence and uniqueness of a nontrivial solution to eigenvalue problems for the following nonlinear fractional differential equation of the form

$$\begin{aligned} \left\{ \begin{array}{l} -D^{\alpha }_{0^{+}}u(t)=\lambda [f(t, u(t), D^{\beta }_{0^{+}}u(t))+g(t)],~~ 0<t<1,\\ u(0)=u(1)=0, \end{array} \right. \end{aligned}$$

where $\lambda$ is a parameter, $D^{\alpha }_{0^{+}},D^{\beta }_{0^{+}}$ are two standard Riemann–Liouville fractional derivatives, $0<\beta <1<\alpha \le 2,\alpha -\beta >1,f: [0,1]\times {\mathbb{R }}\times {\mathbb{R }}\rightarrow {\mathbb{R }}$ is continuous, and $g(t): (0, 1)\rightarrow [0, +\infty )$ is Lebesgue integrable. We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when $\lambda$ is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. In addition, some examples are included to demonstrate the main result.