### The existence of a solution for nonlinear fractional differential equations where nonlinear term depends on the fractional and first order derivative of an unknown function

#### Abstract

In this paper, we consider the existence of solutions of the nonlinear fractional differential equation boundary-value problem

\begin{align*}

&\D_*^{\alpha}u(t)=f(t,u(t),u'(t),{}^C\D^{\beta}u(t)), \quad 0< t<1,\; 1<\alpha< 2,\; 0<\beta\leqslant1,\\

&u(0)=A, \quad u(1)=Bu(\eta),

\end{align*}

where $0<\eta<1$, $A\geqslant 0$, $B\eta>1$, $\D_*^\alpha$ is the modified Caputo fractional derivative of order $\alpha$, ${}^C\D^\beta$ is the Caputo fractional derivative of order $\beta$, and $f$ is a function in $C([0,1]\times\R\times\R\times\R)$.

Existence results for a solution are obtained. Two examples are presented to illustrate the results.

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