Bulletin of the
Korean Mathematical Society BKMS

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) $$\begin{aligned}
-{\bf D}_{0+}^\alpha u(t)=\lambda [f(t,u(t))+q(t)],\quad 0 < t < 1 \\
u(0)=u(1)=0,
\end{aligned}$$ where $\lambda >0$ is a parameter, $1 < \alpha \leq 2$, $\mathbf{D}_{0+}^\alpha$ is the standard Riemann-Liouville differentiation, $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous, and $q(t):(0,1)\to [0, +\infty)$ is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when $\lambda$ in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on $f$.

Keywords: standard Riemann-Liouville differentiation, fractional differential equation, boundary-value problem, nontrivial solution, Leray-Schauder nonlinear alternative

MSC numbers: 34B15, 34B05, 26A33