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On second-order subdifferentials and their applications. (English) Zbl 1011.49016

The authors study second-order subdifferentials of extended-real-valued functions. The starting point is the second-order subdifferential introduced by the first author as the coderivative of the first-order subdifferentiable mapping. Beside this, a semiconvex version is introduced as the coderivative of the convexified first-order subdifferential. Several sum and chain rules are established for both subdifferentials. For a class of separable piecewise \(C^2\) functions, the subdifferentials are efficiently computed.
The theory is then applied to the stability analysis of parametric variational systems of the form \[ 0\in f(x, y)+ Q(x, y), \] where \(f\) is a continuously differentiable vector function and \(Q\) is a multifunction. In this connection, \(y\in \mathbb{R}^m\) is to be interpreted as the decision variable and \(x\in \mathbb{R}^n\) is a perturbation vector. Conditions are established ensuring that the solution map \(S\) defined by \[ S(x):= \{y\in \mathbb{R}^m\mid 0\in f(x, y)+ Q(x,y)\} \] is pseudo-Lipschitzian. Finally, the results are applied to contact problems as well as shape design problems with nonmonotone friction.

MSC:

49J52 Nonsmooth analysis
49J40 Variational inequalities
74M10 Friction in solid mechanics
74M15 Contact in solid mechanics
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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