Asymptotic Behavior of Linear Integrodifferential Systems

Barbu, Viorel; Grossman, Stanley I.

doi:10.2307/1996274

Cited by 10 publications

(13 citation statements)

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“…It follows that S(t) is a C 0 semigroup on Z t . (Compare this semigroup with the one constructed in [2].) THEOREM …”

Section: Preliminaries If X: (-°° 0] -• Cmentioning

confidence: 99%

“…Although semigroup theory is employed in both of the above mentioned articles, it is important to note that the semigroup constructed in [6] differs greatly from the semigroup constructed in [2]. In particular, Miller is able to obtain certain stability results that do not hold for the semigroup constructed in [2]. However, we show that by an appropriate choice of the state space, Miller's semigroup may be considered as the "adjoint" semigroup (in the sense of Hille and Phillips [5]) to the semigroup constructed by Barbu and Grossman.…”

mentioning

confidence: 99%

“…The system under consideration has previously been studied by Barbu and Grossman [2] and Miller [6], via semigroup methods. Although semigroup theory is employed in both of the above mentioned articles, it is important to note that the semigroup constructed in [6] differs greatly from the semigroup constructed in [2]. In particular, Miller is able to obtain certain stability results that do not hold for the semigroup constructed in [2].…”

mentioning

confidence: 99%

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Adjoint semigroup theory for a Volterra integrodifferential system

Burns¹,

Herdman²

1975

Bull. Amer. Math. Soc.

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Communicated by Richard Miller, June 23, 1975 I. Introduction. In this note we announce some recent results concerning the semigroup theory for a class of linear Volterra integrodifferential systems. The system under consideration has previously been studied by Barbu and Grossman [2] and Miller [6], via semigroup methods. Although semigroup theory is employed in both of the above mentioned articles, it is important to note that the semigroup constructed in [6] differs greatly from the semigroup constructed in [2]. In particular, Miller is able to obtain certain stability results that do not hold for the semigroup constructed in [2]. However, we show that by an appropriate choice of the state space, Miller's semigroup may be considered as the "adjoint" semigroup (in the sense of Hille and Phillips [5]) to the semigroup constructed by Barbu and Grossman. We shall state the results without proofs. Proofs of the theorems will appear elsewhere (see [3]).

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“…It follows that S(t) is a C 0 semigroup on Z t . (Compare this semigroup with the one constructed in [2].) THEOREM …”

Section: Preliminaries If X: (-°° 0] -• Cmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Adjoint semigroup theory for a Volterra integrodifferential system

Burns¹,

Herdman²

1975

Bull. Amer. Math. Soc.

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“…Following the construction in [2] for a linear Volterra equation with infinite delay (see Remark 1.5), we define an operator A in X by Then we define A4,** -*. Note that (1.6) implies that * has a strong derivative *' a.e.…”

mentioning

confidence: 99%

“…This space is of "fading memory type" and was studied earlier by B. D. Coleman and V. J. Mizel [6]. Hale uses spectral theory to extend and supplement the results of Barbu and Grossman [2] concerning a linear Volterra integrodifferential equation with infinite delay. This setting was discussed in Remark 1.5.…”

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confidence: 99%

A nonlinear semigroup for a functional differential equation

Brewer¹

1978

Trans. Amer. Math. Soc.

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Abstract. A representation theorem is obtained for solutions of the nonlinear functional differential equationas a semigroup of nonlinear operators on a space of initial data X of "fading memory type." Equation (1) is studied in the abstract setting of a Banach space E. The nonlinear functional F is a uniformly Lipschitz continuous mapping from X to E. The semigroup is constructed by transforming (1) to an abstract Cauchy problemin the space X and applying a generation theorem of M. Crandall and T. Liggett to the operator A in X. The case when (1) is a nonlinear Volterra integrodifferential equation of infinite delay is given special consideration. The semigroup representation is used to obtain finite difference approximations for solutions of (CP) and to study the continuity of solutions of (1) with respect to perturbations of F and $.

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