Nonlocal initial value problems for implicit differential equations with Hilfer–Hadamard fractional derivative

Fractional differential equations (FDEs) have been applied in many fields such as physics, mechanics, chemistry, engineering etc. There has been a significant development in ordinary differential equations involving fractional-order derivatives, one can see the monographs of Hilfer [19], Kilbas [16] and Podlubny [18] and the references therein. Moreover, Hilfer [19] studied applications of a generalized fractional operator having the Riemann– Liouville and the Caputo derivatives as specific cases. Hilfer fractional derivative has been receiving more and more attention in recent times; see, for example, [8–11, 14, 21, 24]. Benchohra et al. [4, 5] studied implicit differential equations(IDEs) of fractional order in various aspects. Recently, some mathematicians have considered FDEs depending on the Hadamard fractional derivative [2, 6, 7].