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Lyapunov functions for evolution variational inequalities with uniformly prox-regular sets. (English) Zbl 1370.34117

This article considers two main topics. The first is studying the regularity of locally absolutely continuous solutions of the differential inclusion \[ \begin{cases} \dot{x}(t,x_0)\in f(x(t,x_0))-N^P(C;x(t,x_0))\text{ a.e }t\in [0,+\infty[,\\ x(t,x_0)\in C\forall t\in[0+\infty[,x(0,x_0)=x_0\quad x_0\in C\end{cases} \] in a Hilbert space, where \(f\) is Lipschitz continuous, \(C\) is a closed \(r\)-uniformly prox-regular set and \(N^{P}(C;x)\) denotes the proximal normal cone at \(x\). The second topic is finding criteria to characterize \(a\)-Lyapunov pairs for (DI) and studying the asymptotic behavior of the solution. The proofs make use of ideas from [S. Marcellin and L. Thibault, J. Convex Anal. 13, No. 2, 385–421 (2006; Zbl 1102.49012)] and [S. Adly et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 3, 985–1008 (2012; Zbl 1239.34070)], among others.

MSC:

34G25 Evolution inclusions
34A60 Ordinary differential inclusions
34D05 Asymptotic properties of solutions to ordinary differential equations
49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
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