## The existence of solutions for boundary value problem of fractional hybrid differential equations.(English)Zbl 1352.34011

Summary: In this paper, we study the existence of solutions for the boundary value problem of fractional hybrid differential equations
$D_{0^+}^\alpha \left[\frac{x(t)}{f(t,x(t))}\right]+g(t,x(t))=0,\quad 0<t<1,\quad x(0)=x(1)=0,$
where $$1<\alpha\leq 2$$ 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.

### MSC:

 34A08 Fractional ordinary differential equations 26A33 Fractional derivatives and integrals 34B15 Nonlinear boundary value problems for ordinary differential equations
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### References:

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