Existence of solutions for a fractional hybrid boundary value problem via measures of noncompactness in Banach algebras

We study the existence of solutions for the following fractional hybrid boundary value problem $$ \cases \displaystyle D_{0^+}^{\alpha}\bigg[\frac{x(t)}{f(t,x(t))}\bigg]+g(t,x(t))=0, &0< t< 1,\\ x(0)=x(1)=0, \endcases $$ where $1< \alpha\leq 2$ and $D_{0^+}^{\alpha}$ denotes the Riemann-Liouville fractional derivative. The main tool is our study is the technique of measures of noncompactness in the Banach algebras. Some examples are presented to illustrate our results. Finally, we compare the results of paper with the ones obtained by other authors