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

d^α₀₊x(t) = f (t, x), t ∈ (0,+∞), 0 < α ≤ 1,

where d^α₀₊ is the standard riemann–liouville fractional derivative, subject to the initial

value condition

lim_t→0+t^1−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,+∞). 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.