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Nonlinear impulsive evolution inclusions of second order. (English) Zbl 1128.34038

The authors consider the second-order nonlinear evolution inclusion with impulses
\[ \ddot u(t)+A(t,\dot u(t))+Bu(t)\in F(t,u(t),\dot u(t)), \]
\[ u(0)=u_0,\;\dot u(0)=v_0, \]
\[ u(t^+_i)\in u(t_i^-)+G_i(u(t_i^-),\dot u(t_i^-)), \]
\[ \dot u(t_i^+)\in\dot u(t_i^-)+H_i(u(t^-_i),\dot u(t_i^-)) \]
in a separable Hilbert space, in which \(F\) and the jump operators \(G_i\) and \(H_i\) are set-valued. Among other hypotheses, \(A\) is assumed to be pseudomonotone and demicontinuous in its second variable, \(B\) is linear, \(F\) is closed-valued, the graph of \(F(t,\cdot, \cdot)\) is sequentially closed, and \(G_i\), \(H_i\) are bounded on bounded sets. Existence of solutions is proven by using a non-impulsive result due to N. S. Papageorgiou and N. Yannakakis, [Acta Mat. Sin., Engl. Ser. 21, No. 5, 977–996 (2005; Zbl 1095.34039)]. Under the additional assumptions that \(F\) is convex-valued and \(G_i,H_i\), have closed graphs, the solution set is proven to be compact.
Two applications are considered in detail: (1) a distributed parameter system governed by a hyperbolic partial differential equation with a feedback control and (2) a mechanical contact problem in viscoelasticity.

MSC:

34G25 Evolution inclusions
35L70 Second-order nonlinear hyperbolic equations
47H04 Set-valued operators
35L15 Initial value problems for second-order hyperbolic equations

Citations:

Zbl 1095.34039
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