The paper is concerned with the existence and uniqueness of a solution to an initial value problem for functional differential equations with unbounded delay and fractional order of the form:
The main tool of proving the results is the nonlinear Leray-Schauder type alternative for contractive mappings in Fréchet spaces due to M. Frigon and A. Granas [Ann. Sci. Math. Qué. 22, No. 2, 161–168 (1998; Zbl 1100.47514)].