The present paper is devoted to establishing the existence result for an integro-differential inclusions governed by a state-dependent nonconvex set-valued mapping \(\mathcal {G}(\cdot )\) in infinite dimensional Hilbert spaces. Under the key assumption \(\mathcal {G}(\mathcal {V})\subset \partial \Phi (\mathcal {V})\) holding on a neighborhood \(\mathcal {V}\) of the initial condition, we develop a comprehensive discretization scheme that addresses all the components of the problem, including the set-valued mapping \(\mathcal {G}(\cdot )\) , to construct a family of approximate solutions converging to the desired solution under some additional mild assumptions. This work improves upon existing results on convex cyclically monotone inclusions in three main directions. First, whereas previous studies were conducted in finite-dimensional spaces \(\mathbb {R}^n\) , our results are established in an infinite-dimensional Hilbert space \(\mathbb {E}\) . Second, while classical assumptions required the hypotheses to hold globally on \(\mathbb {R}^n\) , we only require them to be satisfied locally in a neighborhood of the initial condition \(x_0\) . Furthermore, we relax the usual upper semicontinuity condition by considering the strong-weak case. Additionally, our formulation incorporates further complexity through the presence of a Volterra-type integral term, which significantly broadens the scope of applicability. Finally, we illustrate the applicability of our theoretical findings through an example in electrical circuits with set-valued feedback control.
