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.