Semi-Groups of Operators and Approximation

Butzer, Paul L.; Berens, Hubert

doi:10.1007/978-3-642-46066-1

Cited by 839 publications

(295 citation statements)

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“…Hence, we can apply Theorem 3.2.23 of [3] and obtain that (I 0 + H 0 ) −1 is continuous from (X 0 , D(A 0 )) 1,∞;K to (X 0 , D(A 0 (I 0 + H 0 ))) 1,∞;K which are exactly the Favard classes stated in the thesis. Now, the main result on the growth rate of B is an easy consequence of the previous theorems and propositions.…”

Section: This Proves That (Imentioning

confidence: 60%

Thresholds for macroparasite infections

Tonetto

2004

J. Math. Biol.

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Abstract. We analyse here the equilibria of an infinite system of partial differential equations modelling the dynamics of a population infected by macroparasites. We find that it is possible to define a reproduction number R 0 that satisfies the intuitive definition, and yields a sharp threshold in the behaviour of the system: when R 0 < 1, the parasite-free equilibrium (PFE) is asymptotically stable and there are no endemic equilibria; when R 0 > 1, the PFE is unstable and there exists a unique endemic equilibrium. The results mainly confirm what had been obtained in simplified models, except for the fact that no backwards bifurcation occur in this model. The stability of equilibria is established by extending an abstract linearization principle and by analysing the spectra of appropriate operators.

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Section: This Proves That (Imentioning

confidence: 60%

Thresholds for macroparasite infections

Tonetto

2004

J. Math. Biol.

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“…To emphasize that an operator A generates a semigroup, we will use the standard notation {e tA : t ∈ R + } or {e tA } t∈R + instead of {U(t) : t ∈ R + }. The semigroup {e tA : t ∈ R + } is a C 0 -semigroup iff lim t→+0 e tA x − x = 0 for all x ∈ E. If {e tA } t∈R + is a C 0 -semigroup then the following properties hold (see [3]):…”

Section: Preliminaries and Denotationsmentioning

confidence: 99%

Operator calculus on the class of Sato’s hyperfunctions

Patra¹,

Sharyn²

2013

Carpathian Math. Publ.

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“…Unless otherwise stated, proofs can be found in the book of Hille & Phillips [9]; also see [1,2]. Let X be a (real or complex) Banach space, and let T(t) be a Co-semigroup on X with (infinitesimal) generator A. is entirely contained within the resolvent set p(A) of A.…”

Section: Dual Semigroups Sun-reflexivity and Bounded Perturbationsmentioning

confidence: 99%