Volterra functional differential equations: existence, uniqueness, and continuation of solutions

The initial-value problem for a class of Volterra functional differential equations- of sufficient generality to encompass, as special cases, ordinary differential equations, retarded differential equations, integro-differential equations, and hysteretic differential equations- is studied. A self-contained and elementary treatment of this over-arching problem is provided, in which a unifying theory of existence, uniqueness, and continuation of solutions is developed. As an illustrative example, a controlled differential equation with hysteresis is considered.