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 $s_0(A)$ of the semigroup T is negative provided that all solutions to the Cauchy problems
$$ \dot u(t)=Au(t)+e^{i\mu t}x, \quad t\ge 0,\quad u(0)=0, $$
are bounded (uniformly with respect to the parameter $\mu\in\mathbb{R})$. 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all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid.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 $\mathcal{X}$ denote the space of all continuous and 1-periodic functions $f: \mathbb{B} \to X$ whose sequence of Fourier-Bohr coefficients $(c_m(f))_{m\in\mathbb{Z}}$ belongs to $\ell^1(\mathbb{Z}, X)$. Endowed with the norm $\|f\|_1:=\|(c_m(f))_{m\in\mathbb{Z}}\|_1$ it becomes a non-reflexive Banach space [15]. A subspace $\mathcal{A}_\mathbf{T}$ of X (related to the pair $(\mathbf{T}, \mathcal{X})$ ) 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, $\mathcal{A}_\mathbf{T}=X$. 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 $\mathcal{T}$ associated with T on $\mathcal{X}$ 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 $\mathcal{A}_\mathbf{T}=X$ 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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