Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis

The main aim of this paper is to study the global existence of solutions of initial value problems for nonlinear fractional differential equations(FDEs) on the half-axis, which is fundamental in the basic theory of FDEs and important in stability analysis of this kind of equations. In this paper, we are concerned with the nonlinear FDE D0+αx(t)=f(t,x),t∈(0,+∞),0<α≤1, where D0+α is the standard Riemann–Liouville fractional derivative, subject to the initial value condition limt→0+t1−αx(t)=u0. By constructing a special Banach space and employing fixed-point theorems, some sufficient conditions are obtained to guarantee the global existence of solutions on the interval [0,+∞)[0,+∞). Moreover, in the case α=1α=1, existence results of solutions of initial value problems for ordinary differential equations on the half-axis are also obtained. An interesting example is also included.

► We study the IVPs for nonlinear fractional differential equations.
► We construct a special Banach space.
► Some global existence results of solutions on the half-axis are obtained.
► Existence results of solutions of IVPs for ODEs on the half-axis are also included.