Abstract.

A sufficient condition for the nonexistence of blowing-up mild solutions of a nonlinear evolution fractional functional-differential equation associated with a strongly continuous semigroup and with a nonlinearity containing the Riemann–Liouville fractional integral is established. We prove a result on a new type of nonlinear integral inequalities with weakly singular kernels and delay and apply it in the proof of the result on the nonexistence of blowing-up solutions. This result is applied to a fractionally damped pendulum equation with a time delay forcing term (a feedback control).