A hybrid fractional functional integro-differential equation applies a Riemann-Liouville derivative of order alpha in (0,1) to the quotient of the unknown by a nonlinearity, rather than to the unknown itself, and drives it with a term depending on the solution's entire recent history and on a Volterra integral of that history. Inverting the derivative rewrites the problem as a pointwise product of two operators on the continuous functions with the supremum norm, so a Dhage-type fixed point theorem for products in a Banach algebra applies once one factor is shown Lipschitz and the other compact and continuous by Arzela-Ascoli. The same estimates bound the distance between two solutions by the distance between their initial histories, giving continuous dependence on the initial data with uniqueness as a corollary.
Sidi Ammi, El Kinani and Torres (Electron. J. Differential Equations 2012, no. 103) prove existence and uniqueness for a hybrid functional integro-differential equation of fractional order. The unkno…