In many applications, one assumes the system under consideration is governed by a principle of causality; that is, the future state of the system is independent of the past states and is determined solely by the present. If it is also assumed that the system is governed by an equation involving the state and rate of change of the state, then, generally, one is considering either ordinary or partial differential equations. However, under closer scrutiny, it becomes apparent that the principle of causality is often only a first approximation to the true situation and that a more realistic model would include some of the past states of the system. Also, in some problems it is meaningless not to have dependence on the past. This has been known for some time, but the theory for such systems has been extensively developed only recently. In fact, until the time of Volterra  most of the results obtained during the previous 200 years were concerned with special properties for very special equations. There were some very interesting developments concerning the closure of the set of exponential solutions of linear equations and the expansion of solutions in terms of these special solutions. On the other hand, there seemed to be little concern about a qualitative theory in the same spirit as for ordinary differential equations.
KeywordsFunctional Differential Equation Past State Autonomous Equation Neutral Functional Differential Equation Differential Difference Equation
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