Asymptotic periodicity of recurrent flows in infinite networks

Dorn, Britta; Keicher, Vera; Sikolya, Eszter

doi:10.1007/s00209-008-0410-x

Cited by 15 publications

(10 citation statements)

References 28 publications

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“…During the last decade a deep and extensive theory of network flow semigroups has been developed which deals, among other topics, with the long time behaviour of the flow and relates it to properties of the underlying graph; see e.g. [40,20,19]. However, it seems that so far only positive weights for the mass diversion in the vertices have been considered.…”

Section: The Resolvent Of the Bi-laplace Operator With Dirichlet Bounmentioning

confidence: 99%

Eventually and asymptotically positive semigroups on Banach lattices

Daners

Glück

Kennedy

2016

Journal of Differential Equations

148

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We develop a theory of eventually positive C 0 -semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on L p -spaces, the Dirichlet-to-Neumann operator on L 2 and the Laplacian with non-local boundary conditions on L 2 within the one unified theory. We also introduce and analyse a weaker notion of eventual positivity which we call "asymptotic positivity", where trajectories associated with positive initial data converge to the positive cone in the Banach lattice as t → ∞. This allows us to discuss further examples which do not fall within the above-mentioned framework, among them a network flow with non-positive mass transition and a certain delay differential equation.

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Section: The Resolvent Of the Bi-laplace Operator With Dirichlet Bounmentioning

confidence: 99%

Eventually and asymptotically positive semigroups on Banach lattices

Daners

Glück

Kennedy

2016

Journal of Differential Equations

148

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“…Wang [3] and W.W. Hu [9] separately obtained the exponential stability of series and parallel repairable system by different methods. At the same time, A. Haji [10,14] obtained the well-posedness and stability of the repairable system with primary and secondary failures and the M/M B /1 queueing model by the methods of [15][16][17][18][19] which are completely different from those of [4][5][6][7][8][9][10]. By the graph 6 in the next section, the repairable system could be interpreted as flows of networks.…”

Section: Introductionmentioning

confidence: 98%

“…In recent years, there are abundant results on the well-posedness and stability of the various kinds of repairable systems [3][4][5][6][7][8][9][10] and similar systems, such as queueing systems [11][12][13][14], flows of networks [15][16][17][18][19] and so on [20][21][22]. In 2001, G. Gupur first obtained the well-posedness and asymptotic stability of several queueing system in [11,12] by the c 0 semigroups theory.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and stability of the repairable system with N failure modes and one standby unit

Zheng

Zhu

Gao

2011

Journal of Mathematical Analysis and Applications

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The well-posedness and stability of the repairable system with N failure modes and one standby unit were discussed by applying the c 0 semigroups theory of function analysis. Firstly, the integro-differential equations described the system were transformed into some abstract Cauchy problem of Banach space. Secondly, the system operator generates positive contractive c 0 semigroups T (t) and so the well-posedness of the system was obtained.Finally, the spectral distribution of the system operator was analyzed. It was proven that 0 is strictly dominant eigenvalue of the system operator and the dynamic solution of the system converges to the steady-state solution. The steady-state solution was shown to be the eigenvector of the system operator corresponding to the eigenvalue 0. At the same time the dynamic solution exponentially converges to the steady-state solution.

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“…In this case, the intervals are referred to as edges on which the transport process takes place, while in their endpoints, called vertices , the material is distributed into the adjacent edges. The study of such transport processes using semigroup methods has been initiated under the name flows in networks by 14, and since then, many extensions have been considered, such as 7 or 20. See also 8 for a recent survey article.…”

Section: Introductionmentioning

confidence: 99%

Flows in networks with delay in the vertices

Bayazit

Dorn

Rhandi

2012

Mathematische Nachrichten

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Our goal is to show asynchronous exponential growth (AEG) for a flow in a network with delay in the vertices. For this purpose we show first that its wellposedness can be characterized via an appropriate operator being the generator of a strongly continuous semigroup. We investigate the long term behavior of the system via the spectrum of this generator using techniques from operator matrices, Hille‐Yosida operators and positive semigroups. Finally, we apply our results to deduce that our problem has (AEG).

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Asymptotic periodicity of recurrent flows in infinite networks

Cited by 15 publications

References 28 publications

Eventually and asymptotically positive semigroups on Banach lattices

Eventually and asymptotically positive semigroups on Banach lattices

Well-posedness and stability of the repairable system with N failure modes and one standby unit

Flows in networks with delay in the vertices

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