Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces

Latushkin, Yuri; Montgomery‐Smith, Stephen

doi:10.1006/jfan.1995.1007

Cited by 77 publications

(68 citation statements)

References 25 publications

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“…We prove the theorem in §2 using a recent spectral mapping theorem of Latushkin and Montgomery-Smith ( [7], see Lemma 1 below) and an extrapolation theorem for positive operators on Lp ([11], see Lemma 2 below).…”

Section: Introductionmentioning

confidence: 99%

The stability of positive semigroups on 𝐿_{𝑝} spaces

Weis¹

1995

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

The stability of positive semigroups on 𝐿_{𝑝} spaces

Weis¹

1995

Proc. Amer. Math. Soc.

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“…The proof in [9] used a new spectral mapping theorem for the evolutionary semigroup I ⊗ T t on L q (L p ) by Latushkin and Montgomery-Smith [5] and an extrapolation procedure for the Yosida approximation of T t . In [6] S. Montgomery-Smith simplified the proof by replacing the extrapolation procedure by a direct resolvent estimate.…”

Section: The Resultsmentioning

confidence: 99%

A short proof for the stability theorem for positive semigroups on 𝐿_{𝑝}(𝜇)

Weis

1998

Proc. Amer. Math. Soc.

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Abstract. We give a short proof showing that the growth bound of a positive semigroup on Lp(µ) equals the spectral bound of its generator. It is based on a new boundedness theorem for positive convolution operators on Lp(Lq). We also give a counterexample, showing that Gearhart's result does not extend from Hilbert spaces to Lp(µ)-spaces. The resultsand the growth bound of T t is given by ω(T t ) = inf{ω : ∃C < ∞ with T t ≤ Ce ωt for all t ≥ 0}.The following theorem was proved in [9].The case p = 2 is due to Gearhart and Greiner-Nagel, the case p = 1 is due to Derndinger (see [7], [8], or [3], Theorems 9.5 and 9.7), but the general case remained an open problem for about 10 years. The proof in [9] used a new spectral mapping theorem for the evolutionary semigroup I ⊗ T t on L q (L p ) by Latushkin and Montgomery-Smith [5] and an extrapolation procedure for the Yosida approximation of T t . In [6] S. Montgomery-Smith simplified the proof by replacing the extrapolation procedure by a direct resolvent estimate.In this note we give a new and simpler proof of Theorem 1 that is based on a boundedness result for positive convolutions on mixed norm spaces L p (L q ), and which may be of independent interest (see Theorem 2 below). With this convolution result we can reduce Theorem 1 to a well-known characterization of the spectral bound in terms of weak integrability ([3], Theorem 7.4).Finally, we point out that recent counterexamples concerning stability of semigroups (see e.g. [1]) can be "transplanted" onto L p -spaces. At the end of this note

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“…It is apparent that every linear process with an exponential dichotomy satisfies condition H. More generally, one can show that a linear process {U (t, s) | t ≥ s} having bounded growth satisfies condition H if and only if (see e.g. [25]) there exists an r ∈ (0 , 1) such that the circle with radius r belongs to the resolvent set ρ(T (1)) and that σ(T (1)) ∩ {z ∈ C | |z| < r} = ∅, where {T (t) | t ≥ 0} is the evolution semigroup associated with…”

Section: Definitionmentioning

confidence: 99%

Nonlinear semigroups and the existence and stability of solutionsof semilinear nonautonomous evolution equations

Aulbach

Minh

1996

Abstract and Applied Analysis

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Abstract. This paper is concerned with the existence and stability of solutions of a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the so-called evolution semigroup of (possibly nonlinear) operators. Sufficient conditions for the existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding infinitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes we obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional differential equations.

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Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces

Cited by 77 publications

References 25 publications

The stability of positive semigroups on 𝐿_{𝑝} spaces

The stability of positive semigroups on 𝐿_{𝑝} spaces

A short proof for the stability theorem for positive semigroups on 𝐿_{𝑝}(𝜇)

Nonlinear semigroups and the existence and stability of solutionsof semilinear nonautonomous evolution equations

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