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Unbounded second-order state-dependent Moreau’s sweeping processes in Hilbert spaces. (English) Zbl 1345.34116

The authors consider existence and uniqueness of solutions to the second order sweeping process \[ \begin{aligned} &\ddot u(t)\in-N_{C(t,u(t))}(\dot{u}(t))-F(t,u(t),\dot{u}(t))\text{ a.e. on }[0,T],\\ & u(0)=u_0,\dot{u}(0)\in C\left( 0,u_0\right)\end{aligned} \] in a real Hilbert space \(H\), where \(N_{K}\) represents the normal cone of the set \(K\). The moving set \(C\) is assumed to be nonempty, closed, convex and to satisfy a weak Lipschitz-type assumption and a compactness condition involving the Kuratowski measure of noncompactness. The moving set could be unbounded. \(\;F\) is assumed to be upper semicontinuous with convex, weakly compact values and to satisfy a weak linear growth condition. The specific assumptions are weaker than have been assumed in earlier published results. A second existence result is proven by assuming the moving set is anti-monotone in place of the compactness condition and making a monotonicity assumption on \(F\). The uniqueness result requires a one-sided Lipschitz type assumption on \(F\). The proofs are accomplished with an implicit discretization scheme based on [J. J. Moreau, J. Differ. Equations 26, 347–374 (1977; Zbl 0356.34067)] and uses tools from convex and variational analysis.

MSC:

34G25 Evolution inclusions
49J53 Set-valued and variational analysis

Citations:

Zbl 0356.34067
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References:

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