Exponential stability for solutions of continuous and discrete abstract Cauchy problems in Banach spaces
| dc.contributor.author | Buse, Constantin | |
| dc.contributor.author | Diagana, Toka | |
| dc.contributor.author | Nguyen, Thanh Lan | |
| dc.contributor.author | O'Regan, Donal | |
| dc.date.accessioned | 2021-11-29T17:20:55Z | |
| dc.date.available | 2021-11-29T17:20:55Z | |
| dc.date.issued | 2019-06-04 | |
| dc.description.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all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid.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. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 16 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.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. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/14966 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2019, San Marcos, Texas: Texas State University and University of North Texas. | |
| dc.subject | Uniform exponential stability | |
| dc.subject | Growth bounds for semigroups | |
| dc.subject | Evolution semigroups | |
| dc.subject | Exponentially bounded evolution families of operators | |
| dc.subject | Integral equations in Banach spaces | |
| dc.subject | Fourier series | |
| dc.title | Exponential stability for solutions of continuous and discrete abstract Cauchy problems in Banach spaces | |
| dc.type | Article |