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In this paper we consider so-called locally explicit equations involving nonlinear differentials. Such equations are characterized by certain continuity and semigroup properties of the corresponding quasiflow and arise typically in the mathematical modelling of non-smooth mechanical and physical systems. Under some natural hypotheses, we prove the local solvability of the corresponding Cauchy problem by applying Schauder’s fixed point principle to a suitable equivalent integral equation. Afterwards, we illustrate the abstract existence result by means of an application to an automatic regulation system involving a hysteresis element of stop type.

On the Cauchy Problem for Systems Containing Locally Explicit Equations