Long-time behaviour of solutions to a nonlinear fractional differential system

We consider a subhomogeneous system of two fractional differential equations, which in particular includes a scalar fractional differential equation of order in as a special case. We analyse existence of global positive solutions with given initial conditions and a prescribed long time behaviour. We establish conditions that guarantee the existence of such solutions, and we show that all of them follow precise asymptotic formulas and belong to the class of regularly varying functions, or to a more general class. Some of the results reduce to existing sharp results for second order equations, while others reveal that fractional systems can exhibit behaviour which has no integer order analogy. Several related results are derived, including the global existence of solutions, the necessity of the sufficient conditions, and a possible dichotomy in the solution space.