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

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 s0(A) of the semigroup T is negative provided that all solutions to the Cauchy problems u̇(t) = Au(t) + eiμtx, t ≥ 0, u(0) = 0, are bounded (uniformly with respect to the parameter μ ∈ 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all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid.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 X denote the space of all continuous and 1-periodic functions ƒ : B → X whose sequence of Fourier-Bohr coefficients (cm(ƒ))m∈ℤ∥1 it becomes a non-reflexive Banach space [15]. A subspace AT of X (related to the pair (T, 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, AT = 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 T associated with T on 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 AT = X is not needed in the discrete case.