Initial value problem for nonlinear fractional differential equations with sequential fractional derivative

Abstract

This paper deals with the following initial value problem for nonlinear fractional differential equation with sequential fractional derivative: $$ \left \{ \textstyle\begin{array}{l} {}^{\mathrm{c}}D_{0}^{\alpha_{2}} (\vert {}^{\mathrm{c}}D_{0}^{\alpha_{1}}y(x)\vert ^{p-2} \, {}^{\mathrm{c}}D_{0}^{\alpha_{1}}y(x) )=f(x,y(x)), \quad x>0, y(0)=b_{0},\qquad {}^{\mathrm{c}}D_{0}^{\alpha_{1}}y(0)=b_{1} , \end{array}\displaystyle \right . $$ where ${}^{\mathrm{c}}D_{0}^{\alpha_{1}}$ , ${}^{\mathrm{c}}D_{0}^{\alpha_{2}}$ are Caputo fractional derivatives, $0<\alpha_{1}, \alpha_{2}\le1$ and $p>1$ . We establish the existence and uniqueness of solutions in $C([0,\infty))$ by using the Banach fixed point theorem and an inductive method. An example is presented to illustrate the results in this paper. In addition, existence and uniqueness of solutions of ordinary differential equations with p-Laplacian follow as a special case of our results.