Convergence to logarithmic-type functions of solutions of fractional systems with Caputo-Hadamard and Hadamard fractional derivatives

As a follow-up to the inherent nature of Caputo-Hadamard fractional derivative (CHFD) and the Hadamard fractional derivative ( HFD), little is known about some asymptotic behaviors of solutions. In this paper, a system of fractional differential equations including two types of fractional derivatives the CHFD and the HFD is investigated. The leading derivative is of an order between zero and two whereas the nonlinearities may contain fractional derivatives of an order between zero and two. Under some reasonable conditions, we prove that solutions for the system with nonlinear right hand sides approach a logarithmic function, logarithmic decay and boundedness as time goes to infinity. Our approach is based on a generalized version of Gronwall-Bellman inequality and appropriate desingularization techniques, which we prove. In addition, several manipulations and lemmas such as a fractional version of L'Hopital's rule are used. Our results are illustrated through examples.