## Existence results for evolution hemivariational inequalities of second order.(English)Zbl 1082.34052

The author proves existence of solutions for the second-order nonlinear evolution inclusion $\begin{gathered} y''(t)+ A(t,y'(t))+ By(t)+\partial J(t,y(t))\ni f(t),\quad\text{a.e. on }(0,T),\\ y(0)= y_0,\quad y'(0)= y_1\end{gathered}$ which can be expressed as the hemivariational inequality $\langle y''(t)+ A(t,y'(t))+ By(t)- f(t),v\rangle+ J^0(t, y(t)lv)\geq 0$ for every $$v\in V$$, for almost every $$t\in(0,T)$$ $y(0)= y_0,\quad y'(0)= y_1.$ Here, $$\partial J$$ represents the Clarke subdifferential of $$J$$, $$J(t,x; v)$$ is the generalized directional derivative of $$J(t,\cdot)$$ at a point $$x$$ in the direction $$v$$ and $$V$$ is a reflexive Banach space. This problem can be thought of as an evolution inclusion with a multi-valued perturbation. Physical applications of hemivariational inequalities are mentioned, including references, and a survey of the theoretical results involving these problems is also given. The proof involves an application of the result in [N. S. Papageorgiou, F. Papalini and F. Renzacci, Rend. Circ. Mat. Palermo, II. Ser. 48, No. 2, 341–364 (1999; Zbl 0931.34043)]. Among other hypotheses, $$A$$ is assumed to be pseudomonotone in its second variable, $$B$$ is a bounded, linear, monotone and symmetric operator, and $$J$$ is locally Lipschitz in its second variable. A result is also stated in which the $$\partial J$$ term depends on $$y'$$ rather than $$y$$. Two specific applications from mechanics are given to which the author’s existence result can be applied.

### MSC:

 34G25 Evolution inclusions 35A15 Variational methods applied to PDEs 74H20 Existence of solutions of dynamical problems in solid mechanics 47J20 Variational and other types of inequalities involving nonlinear operators (general)

Zbl 0931.34043
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### References:

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