### Solutions for a hyperbolic system with boundary differential inclusion and nonlinear second-order boundary damping.

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We consider the damped semilinear viscoelastic wave equation $${u}^{\text{'}\text{'}}-\Delta u+{\int}_{0}^{t}h(t-\tau )div\left\{a\nabla u\left(\tau \right)\right\}\mathrm{d}\tau +g\left({u}^{\text{'}}\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{5.0pt}{0ex}}\Omega \times (0,\infty )$$ with nonlocal boundary dissipation. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method provided that the kernel of the memory decays exponentially.

We prove the existence and uniform decay rates of global solutions for a hyperbolic system with a discontinuous and nonlinear multi-valued term and a nonlinear memory source term on the boundary.

In this paper we deal with the anti-periodic boundary value problems with nonlinearity of the form $b\left(u\right)$, where $b\in {L}_{\mathrm{loc}}^{\infty}\left(R\right).$ Extending $b$ to be multivalued we obtain the existence of solutions to hemivariational inequality and variational-hemivariational inequality.

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