On the Nonlinear Fractional Differential Equations with Caputo Sequential Fractional Derivative

Abstract

The purpose of this paper is to investigate the existence of solutions to the following initial value problem for nonlinear fractional differential equation involving Caputo sequential fractional derivativeDc0α2Dc0α1yxp-2Dc0α1yx=fx,yx,x>0,y(0)=b0,Dc0α1y(0)=b1, whereDc0α1,Dc0α2are Caputo fractional derivatives,0<α1,α2≤1,p>1, andb0,b1∈R. Local existence of solutions is established by employing Schauder fixed point theorem. Then a growth condition imposed tofguarantees not only the global existence of solutions on the interval[0,+∞), but also the fact that the intervals of existence of solutions with any fixed initial value can be extended to[0,+∞). Three illustrative examples are also presented. Existence results for initial value problems of ordinary differential equations withp-Laplacian on the half-axis follow as a special case of our results.