Integrated semigroups, C-semigroups and the abstract Cauchy problem

deLaubenfels, Ralph

doi:10.1007/bf02573380

Cited by 54 publications

(33 citation statements)

References 15 publications

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“…A generalization of strongly continuous semigroups, regularized semigroups (Definition 2.1) was introduced, independently, by DA PRATO [4] and DAVIES and PANG [6], and extended in [7]; see also [22], [23], [27], [8] and [9]. Wide classes of operators, A, for which (1.1) may be dealt with using regularized semigroups, but not strongly continuous semigroups, appear in [7], [8], [9] and [10].…”

Section: Du (T) = A(u(t)) (T ~ O) U(o) = Xmentioning

confidence: 99%

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Regularized semigroups, iterated Cauchy problems and equipartition of energy

Goldstein

Laubenfels

SandefurJr.

1993

Monatshefte für Mathematik

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Abstract. The iterated Cauchy problem under consideration is l~nk = 1 (did t -Ak) u(t) = 0 (t >1 0). (*)Here {A 1 ..... An} are unbounded linear operators on-a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When each A k is a (Co) semigroup generator, this semigroup is of class (Co) and was studied by J. T. Sandefur [26]. This result is extended to the case when each A k generates a C-regularized semigroup (with C independent of k). This means one can solve u' = Au, u (0) = fe C (Dora (A)) and get u(t)~ 0 whenever C-l f--* 0; here C is bounded and injective. When the A k are commuting generator with A k -Aj injective for k #j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form u = ~= 1 ui where ui is a solution of u; = A i ui. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is weUknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy.

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Section: Du (T) = A(u(t)) (T ~ O) U(o) = Xmentioning

confidence: 99%

“…Wide classes of operators, A, for which (1.1) may be dealt with using regularized semigroups, but not strongly continuous semigroups, appear in [7], [8], [9] and [10].…”

Section: Du (T) = A(u(t)) (T ~ O) U(o) = Xmentioning

confidence: 99%

Regularized semigroups, iterated Cauchy problems and equipartition of energy

Goldstein

Laubenfels

SandefurJr.

1993

Monatshefte für Mathematik

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“…The rest can be proved in [2,3]. After the paper w accepted, the author understood that the equivalence of (a) and (b) h been extended to the ce D(A) X by Shaw and Li [7].…”

Section: Proposition 2 Let (Ac(t)) G(mwx) (K)mentioning

confidence: 99%

Integrated cosine functions

Zheng

1994

International Journal of Mathematics and Mathematical Sciences

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“…In the theory of semigroups of operators, it is known that whether a linear operator A is the generator of a certain semigroup (C 0 -semigroup or integrated semigroup) is related to the Laplace representation of its resolvent R(λ, A) (see [1], [5], [3]). In Section 3, using the results in Section 2, we obtain some characterization results for generators of semigroups of operators.…”

Section: Introductionmentioning

confidence: 99%

Laplace transforms and generators of semigroups of operators

Peng

Chung

1998

Proc. Amer. Math. Soc.

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Abstract. In this paper, a characterization for continuous functions on (0, ∞) to be the Laplace transforms of f ∈ L ∞ (0, ∞) is obtained. It is also shown that the vector-valued version of this characterization holds if and only if the underlying Banach space has the Radon-Nikodým property. Using these characterizations, some results, different from that of the Hille-Yosida theorem, on generators of semigroups of operators are obtained.

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Integrated semigroups, C-semigroups and the abstract Cauchy problem

Cited by 54 publications

References 15 publications

Regularized semigroups, iterated Cauchy problems and equipartition of energy

Regularized semigroups, iterated Cauchy problems and equipartition of energy

Integrated cosine functions

Laplace transforms and generators of semigroups of operators

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