Abstract Volterra Equations: A Survey

The aim of this paper is to summarize some of the basic concepts and results, available in the mathematical (and partially) or engineering literature, related to functional or differential functional equations with abstract Volterra operators (causal or nonanticipative, as these operators are called in the engineering literature). Equations such as x(t) = (Vx)(t), x′(t) = (Vx)(t), or (Vx)(t) = (Wx)(t), where V and W are causal operators, are dealt with. Existence, uniqueness of solutions in various function spaces are established. Further results on boundary value problems, stability, or asymptotic behavior are considered. Connections with classical types of functional equations are emphasized. Some applications are briefly indicated.