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

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 View the MathML sourceD0+αx(t)=f(t,x),t∈(0,+∞),0<α≤1, Turn MathJax on where View the MathML sourceD0+α is the standard Riemann–Liouville fractional derivative, subject to the initial value condition View the MathML sourcelimt→0+t1−αx(t)=u0. Turn MathJax on 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.