We have found a unified method to analyse Timoshenko beams under various boundary conditions that occurred in practice. Explicit asymptotic expressions for the spectrum are obtained. Our method is very simple but effective because explicit formulas are obtained for the system of fundamental solutions, which are very useful for other purposes such as stability analysis. The eigenfunctions are also shown to form an orthogonal basis.
Given matrices A i , B i and C i (i ∈ I) of corresponding dimensions over a field F, we prove that:then there exists a simultaneous solution X to the matrix Sylvester equations A i X − XB i = C i ; and (ii) if ⎛ ⎝ A i C i O B i ⎞ ⎠ are simultaneously equivalent to ⎛ ⎝ A i O O B i ⎞ ⎠ , then there exist simultaneous solutions X, Y to the matrix equations A i XWe also show that analogous results hold for mixed pairs of matrix Sylvester equations A 1 X 1 − YB 1 = C 1 , A 2 X 2 − YB 2 = C 2 and for generalized Stein equations X − AYB = C.
The stability of solutions of the equation Bu 0 (t) = Au(t) is considered, where A and B are closed linear operators on a Banach space. Under the well-posedness condition it is proved that if the imaginary part of the spectrum of the pencil (¸B ¡ A) is countable, then a bounded uniformly continuous solution u(t) of the equation is asymptotically almost periodic if and only if the functions e ¸tu (t), (¸2 iR), have uniformly convergent means. A condition of exponential stability also is given when the generalized eigenvectors and associated root vectors of the linear pencil (¸B ¡ A) form a Riesz basis.
A discrete analog of a construction of the Hille Yosida space is used to obtain results on the asymptotic behaviour of individual orbits of generally unbounded operators. Results on power-bounded linear operators on a Banach space are applied to the restriction of the operator on the Hille Yosida space. The same method is used to obtain an individual variant of the Katznelson Tzafriri Theorem. Continuous analogs for some of these results will be presented elsewhere.
1996Academic Press, Inc.
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