Abstract integro-differential equations with state-dependent integration intervals: Existence, uniqueness, and local well-posedness

In this work, we study a new class of integro-differential equations with delay, where the informations from the past are represented as an average of the state over state-dependent integration intervals. We establish results on the local and global existence and qualitative properties of solutions. The models presented and the ideas developed will allow the generalization of an extensive literature on different classes of functional differential equations. The last section presents some examples motivated by integro-differential equations arising in the theory of population dynamics.