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

Abstract

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 D 0 + α x ( t ) = f ( t , x ) , t ∈ ( 0 , + ∞ ) , 0 < α ≤ 1 , where D 0 + α is the standard Riemann–Liouville fractional derivative, subject to the initial value condition lim t → 0 + t 1 − α x ( t ) = u 0 . 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 , + ∞ ) . Moreover, in the case α = 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.