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In this paper, we develop the Galerkin-like method to address 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 provide newinsights for differential inclusions with an unbounded right-hand side. Theeffectiveness of the proposed approach is illustrated by presenting newexistence results for nonconvex state-dependent Volterra sweeping processes,where the right-hand side is unbounded, and the classical theory ofdifferential inclusions is not applicable. This is the first result of itskind.  The paper concludes with an application to the existence of an optimalcontrol problem governed by nonconvex state-dependent Volterra sweepingprocesses in finite dimensions.

Galerkin-like Method for Integro-Differential Inclusions with applications to Volterra Sweeping processes