Constantin Buse, Toka Diagana, Lan Thanh Nguyen, Donal O'Regan

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all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid.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 | |

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 | |

Lan Thanh Nguyen Western Kentucky University Department of Mathematics Bowling Green, KY 42101, USA email: lan.nguyen@wku.edu | |

Donal O'Regan National University of Ireland Galway, Ireland email: donal.oregan@nuigalway.ie |

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