Abstract
Current paper deals with another type of generalized Timoshenko beam equations, we consider coefficients - in \(L_\infty {[0,1]}\), depending on the spatial variable only. We will establish a sufficiently conditions that the differential operator on the space - defines a \(C_0\) - analytic semigroup of contractions, which actually is stable (Lyapunov sense) on time, however it defines completely periodic solutions. We have already studied the important cantilever case, see [17], now we even will improve the estimation and to show that in the clamped case, more severe conditions on the elastic medium are to be imposed without causing instability in the dynamics of the regarded system. In the same fashion, the whole spectrum of the regarded operator consists only of completely imaginary isolated points with a unique accumulation point.
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Notes
- 1.
In the case G(1, 0), obviously we obtain that \(\forall t\in (0,+\infty ]\), \(T_{t}\) is a 1-Lipschitz operator, in other words, it is a contractive map, henceforth we call such a family a semigroup of contractions, with \(T_{0}=Id_{\textbf{H}}, \) moreover it is a norm-strongly continuous in its parameter from the right of 0..
- 2.
The closure of the finite rank operators in the operator norm over the natural Banach algebra of the continuous endomorphisms over X is the (maximal) two-sided topological ideal \(\mathcal {K}\) of the compact linear operators over X, i.e. X admits a Schauder Basis, i.e. for instance \(L^{2}[0,L].\).
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Acknowledgements
The authors thanks to his teacher Professor O. Kalenda and Doc. T. Barta for their valuable consultations regarding the general theory of closed and densely-defined unbounded linear operators and the semigroup of contractions. As well, it is worth mentioning the role of whole Department of Mathematical Analysis - at Charles University in Prague, Faculty of Mathematics and Physics.
Moreover, the author was supported by the institutional support, grant SVV-2020-260583 and SVV-2023-260711.
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Popov, S.I. (2023). The \(c_{0}-\) Analytic Semigroup of Contractions, Generated by a Kind of Timoshenko Beam Type Dirichlet Boundary Operator. In: Georgiev, I., Kostadinov, H., Lilkova, E. (eds) Advanced Computing in Industrial Mathematics. BGSIAM 2019. Studies in Computational Intelligence, vol 1111. Springer, Cham. https://doi.org/10.1007/978-3-031-42010-8_18
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