Electron. J. Differential Equations,
Vol. 2019 (2019), No. 78, pp. 1-16.
Exponential stability for solutions of continuous and
discrete abstract Cauchy problems in Banach spaces
Constantin Buse, Toka Diagana, Lan Thanh Nguyen, Donal O'Regan
Abstract:
Let T be a strongly continuous semigroup acting on a complex
Banach space X and let A be its infinitesimal generator.
It is well-known [29,33] that the uniform spectral bound
of the semigroup T is negative provided that all solutions to
the Cauchy problems

are bounded (uniformly with respect to the parameter
.
In particular, if X is a Hilbert space, then this yields all trajectories
of the semigroup T are exponentially stable, but if X is an
arbitrary Banach space this result is no longer valid. Let
denote the space of all continuous and 1-periodic functions
whose sequence of Fourier-Bohr coefficients
belongs to
.
Endowed with the norm
it becomes
a non-reflexive Banach space [15].
A subspace
of X (related to the pair
) is introduced in the third section of this paper.
We prove that the semigroup T is uniformly exponentially stable
provided that in addition to the above-mentioned boundedness condition,
.
An example of a strongly continuous semigroup (which is not uniformly continuous)
and fulfills the second assumption above is also provided. Moreover an
extension of the above result from semigroups to 1-periodic and strongly
continuous evolution families acting in a Banach space is given.
We also prove that the evolution semigroup
associated with T on
does not verify the spectral determined growth
condition. More precisely, an example of such a semigroup with uniform spectral
bound negative and uniformly growth bound non-negative is given.
Finally we prove that the assumption
is not
needed in the discrete case.
Submitted December 19, 2018. Published June 4, 2019.
Math Subject Classifications: 35B35, 47A30, 46A30.
Key Words: Uniform exponential stability; growth bounds for semigroups;
evolution semigroups; exponentially bounded evolution families of operators;
Integral equations in Banach spaces; Fourier series.
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Constantin Buse
Politehnica University of Timisoara
Department of Mathematics, Piata Victoriei No. 2
300006-Timisoara, Romania
email: constantin.buse@upt.ro
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Toka Diagana
University of Alabama in Huntsville
Department of Mathematical Sciences
301 Sparkman Drive, Shelby Center
Huntsville, AL 35899, USA
email: toka.diagana@uah.edu
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Lan Thanh Nguyen
Western Kentucky University
Department of Mathematics
Bowling Green, KY 42101, USA
email: lan.nguyen@wku.edu
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Donal O'Regan
National University of Ireland
Galway, Ireland
email: donal.oregan@nuigalway.ie
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