  In this article we study the basic theoretical properties of Mellin-type
fractional integrals, known as generalizations of the Hadamard-type fractional
integrals. We give a new approach and version, specifying their semigroup
property, their domain and range. Moreover we introduce a notion of strong
fractional Mellin derivatives and we study the connections with the pointwise
fractional Mellin derivative, which is defined by means of Hadamard-type
fractional integrals. One of the main results is a fractional version of the
fundamental theorem of differential and integral calculus in the Mellin frame.
In fact, in this article it will be shown that the very foundations of Mellin
transform theory and the corresponding analysis are quite different to those of
the Fourier transform, alone since even in the simplest non-fractional case the
integral operator (i.e. the anti-differentiation operator) applied to a
function f will turn out to be the $R \int_0^x f(u)du=u$ with derivative
$(xd/dx)f(x)$. Thus the fundamental theorem in the Mellin sense is valid in
this form, one which stands apart from the classical Newtonian integral and
derivative. Among the applications two fractional order partial differential
equations are studied.
