Asymptotic behavior of solutions of fractional differential equations with hadamard fractional derivatives

The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l’Hopital’s rule which we prove here.