Measure functional differential equations with infinite time-dependent delay

In this work, we introduce the measure functional differential equations (MFDEs) with infinite time-dependent delay, and we study the correspondence between the solutions of these equations and the solutions of the generalized ordinary differential equations (GODEs, for short) in Banach spaces. Using the theory of GODEs, we obtain results concerning the existence and uniqueness of solutions and continuous dependence on parameters for MFDEs with infinite time-dependent delay. We develop the theory in the context of phase spaces defined axiomatically. Our results in this paper generalize several previous works on MFDEs with infinite time-independent delay.