This paper deals with fractional calculus. The fractional primitive of order of a function is given by
provided the right-side is pointwise defined on . The fractional derivative of order of a continuous function is given by
provided the right-side is pointwise defined on .
The authors consider the fractional differential equation
where and , , is a given function, continuous in . Under some assumptions, equation (1) is equivalent to the integral equation , reduction used systematically in this paper.
A real-valued function , or in the case , with fractional derivative on , is a solution of (1) if for all .
The authors prove that if , is a continuous function in and is continuous on , then (1) has at least one continuous solution on for a suitable . Then, the authors show that uniqueness and global existence of solutions of (1) can be obtained a uniform Lipschitz-type assumption.
The last section of the paper concerns initial value problems of the type (1) and with and .