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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.

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.

Reviewer: W.Schirotzek (Dresden)

### 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 |