Lyapunov-type inequality and positive solutions for a nonlinear fractional boundary value problem

In this paper, we extend our exploration of a fractional boundary value problem involving left Riemann–Liouville and right Caputo fractional derivatives with order \(\frac{1}{2}<\alpha \le 1,\) $\frac{1}{2}<\alpha \le 1,$ as in [13]. Firstly, we investigate the properties of associated Green’s function. Secondly, we prove the existence of positive solutions, where the Guo-Krasnoselskii fixed-point theorem and properties of the Green function play key roles in the proof. Also, we show the generalised Lyapunov and Hartman–Wintner-type inequalities for the associated fractional boundary value problem. Finally, we present some examples for fractional boundary value problems.