The existence of solutions for boundary value problem of fractional hybrid differential equations

In this paper, we study the existence of solutions for the boundary value problem of fractional hybrid differential equations$D_{0^{+}}^{\alpha }\left[\frac{x(t)}{f(t\text{,}x(t))}\right]+g(t\text{,}x(t))=0\text{,}0<t<1\text{,}$$x(0)=x(1)=0\text{,}$where $1<\alpha ?2$ is a real number, $D_{0^{+}}^{\alpha }$ is the Riemann–Liouville fractional derivative. By a fixed point theorem in Banach algebra due to Dhage, an existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. As an application, examples are presented to illustrate the main results.