We deeply investigate the well-posedness of models taking the form $_0^AD^{\beta }_tu(t) = Au(t),\;\; u(0)= \,f,\;\;\;00$ where $_0^AD^{\beta }_t$ is a derivative with the fractional parameter $\beta$ and $A$ is a closed densely defined operator in a Banach space. We show that, unlike other systems, solutions of our models are not governed by Mittag--Leffler functions and their variants. We extend and adapt Peano's idea to our models and establish conditions for existence and uniqueness of solutions. In particular, relations between the two-parameter solution operator, its resolvent, and its generator are provided; the issue of subordination and prolongation principles are addressed; and a way to approximate the generalized solution is presented. Finally, application to transport-convection differential equations is performed in the space of distributions with finite higher moments to show how their well-posedness can be addressed.