The Convergence Problem for Dissipative Autonomous Systems by Alain Haraux & Mohamed Ali Jendoubi

Author:Alain Haraux & Mohamed Ali Jendoubi
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

7.4 A Class of Second Order Systems

Let be as in Sect. 7.3. We consider the equation

(7.8)

Corollary 7.4.1

Any solution u(t) of (7.8) defined and bounded on together with satisfies

In other terms we have In addition, if for each c, the set is discrete, then there exists such that

Proof

We consider the dynamical system generated by (7.8) on the closure of the range of . Here the set of fixed points of S(t) is made of points for which the solution u of (7.8) of initial data (y, z) is independent of t. Consequently . Multiplying by in the sense of the inner product of and integrating we find

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