  In this paper, we analyze and discuss the well-posedness of a new variant of
the so-called sweeping process, introduced by J.J. Moreau in the early 70's
\cite{More71} with motivation in plasticity theory. In this variant, the normal
cone to the (mildly non-convex) prox-regular moving set $C(t)$, supposed to
have an absolutely continuous variation, is perturbed by a sum of a
Carath\'{e}odory mapping and an integral forcing term. The integrand of the
forcing term depends on two time-variables, that is, we study a general
integro-differential sweeping process of Volterra type. By setting up an
appropriate semi-discretization method combined with a new Gronwall-like
inequality (differential inequality), we show that the integro-differential
sweeping process has one and only one absolutely continuous solution. We also
establish the continuity of the solution with respect to the initial value. The
results of the paper are applied to the study of nonlinear integro-differential
complementarity systems which are combination of Volterra integro-differential
equations with nonlinear complementarity constraints. Another application is
concerned with non-regular electrical circuits containing time-varying
capacitors and nonsmooth electronic device like diodes. Both applications
represent an additional novelty of our paper.
