Well-posedness for integro-differential sweeping processes of Volterra type (Q6620714)

For a real Hilbert space \(H\), the \textit{J. J. Moreau}'s sweeping process [Trav. Semin. d'Anal. convexe, Montpellier 1, Exposé No. 15, 43 p. (1971; Zbl 0343.49019); ibid. 2, Exposé No. 3, 36 p. (1972; Zbl 0343.49020)] is a differential inclusion involving the normal cone to a family of closed moving sets. In this paper, the author is interested in a new variant of the sweeping process called the integro-differential sweeping process of Volterra type. In its general form, it is the following differential inclusion:\N\[\N\dot{x}(t)\in-N_{C(t)}(x(t))+f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds\qquad\text{ a.e. }t \in I \tag{\(*\)}\N\]\N\(x(T_{0}) = x_{0} \in C(T_{0})\), \(T_{0}\in I\), where \(C(t)\) is a non-empty, closed and \(\rho\)-uniformly prox-regular set of all \(t\in I\) and \(f:I\times H \rightarrow H\) and \(g:I\times I \rightarrow H\) are functions satisfying certain conditions mentioned in the paper. The integro-differential sweeping process \((*)\) is a natural extension of the sweeping process.\N\NIt has been shown, in this paper, that well-posedness for the integro-differential sweeping process can be obtained through a reparametrization technique and a fixed point argument for history-dependent operators. Moreover, a fully continuous dependence result with respect to the data of the problem is presented. This approach is based on some enhanced version of the Gronwall's inequality. The paper ends with an application to projected dynamical systems.