Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equation boundary-value problem $$\left\{ \begin{array}{@{}l}-D^{\alpha}_{0+}u(t)=f(t,u(t)), \quad t\in[0,1]\\[3pt]u(0)=u(1)=u''(0)=0\end{array} \right.$$ where 2<α≤3 is a real number, and \(D^{\alpha}_{0+}\) is the Caputo’s fractional derivative, and f:[0,1]×[0,+∞)→[0,+∞) is continuous. By means of a fixed-point theorem on cones, some existence, nonexistence and multiplicity of positive solutions are obtained.