An extension of the well-posedness concept for fractional differential equations of Caputo’s type

Abstract

It is well known that, under standard assumptions, initial value problems for fractional ordinary differential equations involving Caputo-type derivatives are well posed in the sense that a unique solution exists and that this solution continuously depends on the given function, the initial value, and the order of the derivative. Here, we extend this well-posedness concept to the extent that we also allow the location of the starting point of the differential operator to be changed, and we prove that the solution depends on this parameter in a continuous way too if the usual assumptions are satisfied. Similarly, the solution to the corresponding terminal value problems depends on the location of the starting point and of the terminal point in a continuous way too.

Keywords:

• fractional differential equation
• Caputo derivative
• well-posed problem
• continuous dependence
• initial value
• terminal value
• starting point

AMS Subject Classifications:

• 34A08
• 34A12
• 26A33

Notes

Dedicated to my teacher Professor Dr. Heinz-Wilhelm Alten on the occasion of his 85th birthday.