8Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 78, pp. 1-16.

Title: Exponential stability for solutions of continuous and
       discrete abstract Cauchy problems in Banach spaces

Authors: Constantin Buse (Polytechnic Univ. of Timisoara, Romania)
         Toka Diagana (Univ. of Alabama in Huntsville, AL, USA)
         Lan Thanh Nguyen (Western Kentucky Univ., Bowling Green, KY, USA)
         Donal O'Regan (National Univ. of Ireland, Galway, Ireland)

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 arbitrary Banach space this result is no longer valid.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 04, 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.