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On a class of eigenvalue problems and their applications. (English) Zbl 1123.34306

The authors consider the boundary value problem for the system of ordinary differential equations
\[ K(w''-\varphi')+\omega^2\rho w=0, \]
\[ EI\varphi''-K(\varphi-w')+\omega^2 I_{\rho}\varphi=0, \]
\[ \mathcal{A} J(0)=0,\qquad \mathcal{B} J(1)=0, \]
where \(\mathcal{A}\) and \(\mathcal{B}\) are \(4\times 4\) real matrices, \(J(x)=(w(x),w'(x),\varphi(x),\varphi'(x))^T\), \(\rho\), \(I_{\rho}\), \(K\), \(EI\) are positive real numbers, and \(i\omega\) is a nonzero eigenvalue associated to certain distributed system. The authors derive conditions guaranteeing the nonexistence of a nontrivial solution to the problem considered. The obtained results are applied to determine the asymptotic behaviour of a certain class of Timoshenko beam with dissipative boundary feedback.
Reviewer: Robert Hakl (Brno)

MSC:

34B05 Linear boundary value problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34C60 Qualitative investigation and simulation of ordinary differential equation models
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