A new Grünwald–Letnikov operator using a continuous positive function

Abstract

UDC 517.5; 517.9

We propose a Grünwald–Letnikov-type derivative using a continuous positive function $T.$ This function generalizes some conformable and nonconformable derivative kernels known from the literature. First, we obtain basic properties, such as the Leibniz rule and the index law for this fractional derivative. We also prove integral representations and show that this derivative does not, in general, map periodic functions onto periodic functions with the same period. Finally, we conclude that periodic functions cannot be solutions of certain differential equations involving this Grünwald–Letnikov-type derivative for particular values of the order.