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

#### Abstract

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.

$$

\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.

#### Keywords

Banach algebras; Riemann-Liouville fractional derivative; measure of noncompactness; hybrid boundary value problem

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