State dependent delay differential equations with distributed memory

We investigate state dependent delay differential equations with distributed memory, combining discrete state dependent delays and a convolution type memory operator. Under Lipschitz type assumptions on the delay, kernel, and nonlinear term, we establish local well posedness using a fixed point argument in a Banach space of histories. The distributed memory operator is shown to be locally Lipschitz, ensuring existence and uniqueness. As a benchmark, we analyze a logistic model with a Gamma distributed memory kernel of order two. The convolution term is reduced to a finite dimensional form, yielding an explicit characteristic equation and closed form Hopf bifurcation threshold and frequency. The results show that higher order memory kernels may induce oscillatory dynamics, unlike purely exponential memory, and provide a rigorous framework for stability and bifurcation analysis in delay systems with state dependent memory.