Displaying similar documents to “On the completeness of the system { t λ n log m n t } in C 0 ( E )

Remarks on the uniqueness of second order ODEs

Dalibor Pražák (2011)

Applications of Mathematics

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We are concerned with the uniqueness problem for solutions to the second order ODE of the form x ' ' + f ( x , t ) = 0 , subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x , for each t fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems...

Growth orders of Cesàro and Abel means of uniformly continuous operator semi-groups and cosine functions

Ryotaro Sato (2010)

Commentationes Mathematicae Universitatis Carolinae

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It will be proved that if N is a bounded nilpotent operator on a Banach space X of order k + 1 , where k 1 is an integer, then the γ -th order Cesàro mean C t γ : = γ t - γ 0 t ( t - s ) γ - 1 T ( s ) d s and Abel mean A λ : = λ 0 e - λ s T ( s ) d s of the uniformly continuous semigroup ( T ( t ) ) t 0 of bounded linear operators on X generated by i a I + N , where 0 a , satisfy that (a) C t γ t k - γ ( t ) for all 0 < γ k + 1 ; (b) C t γ t - 1 ( t ) for all γ k + 1 ; (c) A λ λ ( λ 0 ) . A similar result will be also proved for the uniformly continuous cosine function ( C ( t ) ) t 0 of bounded linear operators on X generated by ( i a I + N ) 2 .

Two notes on eventually differentiable families of operators

Tomáš Bárta (2010)

Commentationes Mathematicae Universitatis Carolinae

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In the first note we show for a strongly continuous family of operators ( T ( t ) ) t 0 that if every orbit t T ( t ) x is differentiable for t > t x , then all orbits are differentiable for t > t 0 with t 0 independent of x . In the second note we give an example of an eventually differentiable semigroup which is not differentiable on the same interval in the operator norm topology.