On the fractional solution of the equation f(x + y) = f(x)f(y) and its application to fractional Laplace’s transform

It is shown that, if the problem is defined in the setting of fractional calculus via fractional difference on non-differentiable functions, then the solution of the functional equation f(x + y) = f(x)f(y) is exactly defined as the solution of a linear fractional differential equation. The dual or counterpart problem, that is the fractional solution of the equation g(xy) = g(x) + g(y), is also considered, and it is shown that the corresponding solution is the logarithm of fractional order defined as the inverse of a generalized Mittag–Leffler function which is nowhere differentiable. This framework suggests a definition of fractional Laplace’s transform expressed in terms of generalized Mittag–Leffler function, and its main properties are outlined: mainly inverse function and convolution. One takes this opportunity to display a (new) fractional Taylor’s series for functions f(x,y) of two variables x and y. Many open problems are stated, which are directly related to the non-differentiability of the functions so involved, therefore the title “on the fractional solution…”.