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                <term xml:lang="en">velocity constraint</term>
                <term xml:lang="en">local Lipschitz-likeness</term>
                <term xml:lang="en">Bochner integration</term>
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                <term xml:lang="en">uniform prox-regularity</term>
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                <term xml:lang="en">47H04</term>
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              <p>In this paper, we study some classes of sweeping processes with velocity constraints in the moving set. In addition to the solution existence and the solution uniqueness for the case of a moving convex constraint set, some results on the solution existence and the solution multiplicity where the constraint set is a finite union of disjoint convex sets are also obtained. Our main tool is a theorem on the solution sensitivity of parametric variational inequalities. Beside the traditional requirement that the constraint set moves continuously in the Hausdorff distance sense, we intensively use a new assumption on the local Lipschitz-likeness of the constraint set-valued mapping. The obtained results are compared with the existing ones and analyzed by several examples.</p>
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