I am looking for the solution of $$\frac{d^\alpha}{d x^\alpha}f(x)=g(x)f(x),$$ where $\alpha \in (0,1)$ and $\frac{d^\alpha}{d x^\alpha}$ is the Caputo derivative.
A series of Jumarie's papers, "2005On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion", "2005On the representation of fractional Brownian motion as an integral with respect to and $(dt)^\alpha$" and "2006Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results", and many other later papers, the author claim the solution is $$ f(x)=f(0)E_\alpha\left(\alpha \int_0^x (x-y)^{\alpha-1}g(y)dy\right), $$ where $E_\alpha$ is the Mittag-Leffler function. Apart from the idea provided in these papers, do we have other ways to prove it?
By the way, the author also apply one result of the Mittag-Leffler function, $$ E_\alpha\left((x+y)^\alpha\right)=E_\alpha(x^\alpha)E_\alpha(y^\alpha). $$ May I ask what is the condition for this to be true? At least, Wolfram Mathematica does not support this equation.
Many thanks in advance.