Exponential stability for solutions of continuous and discrete abstract Cauchy problems in Banach spaces
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Date
2019-06-04
Authors
Buse, Constantin
Diagana, Toka
Nguyen, Thanh Lan
O'Regan, Donal
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid.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.
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Keywords
Uniform exponential stability, Growth bounds for semigroups, Evolution semigroups, Exponentially bounded evolution families of operators, Integral equations in Banach spaces, Fourier series
Citation
Buse, C., Diagana, T., Nguyen, L. T., & O'Regan, D. (2019). Exponential stability for solutions of continuous and discrete abstract Cauchy problems in Banach spaces. <i>Electronic Journal of Differential Equations, 2019</i>(78), pp. 1-16.
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Attribution 4.0 International