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.