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In this paper, we develop the Galerkin-like method to deal with first-orderintegro-differential inclusions. Under compactness or monotonicity conditions,we obtain new results for the existence of solutions for this class ofproblems, which generalize existing results in the literature and give newinsights for differential inclusions with an unbounded right-hand side. Theeffectiveness of the proposed approach is illustrated by providing new resultsfor nonconvex state-dependent integro-differential sweeping processes, wherethe right-hand side is unbounded, and the classical theory of differentialinclusions is not applicable. It is the first result of this kind. The paperends with an application to the existence of an optimal control problemgoverned by an integro-differential inclusion in finite dimensions.

Galerkin-like method for integro-differential inclusions with application to state-dependent sweeping processes