Semigroups of operators, cosine operator functions, and linear differential equations

Vasil'ev, V. V.; Крейн, С. Г.; Piskarev, Sergey

doi:10.1007/bf01138948

Cited by 51 publications

(27 citation statements)

References 182 publications

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“…It is well known that paralleling the theory of semigroups of operators, one can develop the corresponding theory of cosine functions Ž w x. see, e.g., 29 . Recently, the theory of regularized cosine functions has w x w x been developed by Kuo On the other hand, the almost periodicity of C -semigroups and C -0 0 w x groups have been studied systematically by Bart and Goldberg 3 .…”

Section: Introductionmentioning

confidence: 99%

Almost Periodic Regularized Groups, Semigroups, and Cosine Functions

Zheng

Liu

1996

Journal of Mathematical Analysis and Applications

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Section: Introductionmentioning

confidence: 99%

Almost Periodic Regularized Groups, Semigroups, and Cosine Functions

Zheng

Liu

1996

Journal of Mathematical Analysis and Applications

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“…If Al, A2 E Go, then, in general, the operator A1 + A2 (or its closure) does not belong to Go even if C(t, A1) commutes with C(t, A2) (see [1]). However, the following assertion is valid.…”

Section: (3)mentioning

confidence: 97%

“…The function Yt(t, A) will be referred to as an operator Beasel function; the set of operators for which problem (1), (2) is uniformly correct will be denoted by Gt.…”

Section: (3)mentioning

confidence: 99%

Perturbations of an abstract Euler-Poisson-Darboux equation

Glushak

1996

Math Notes

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ABSTRACT. The stability of the uniform correctness of the Cauchy problem u"(t) + ~u'(t) = Au(t), t > 0, u(0) = u0, u'(0) = O, for k > 0 with respect to perturbations of the operator A is studied.KEY WORDS: abstract Cauchy problem, uniform correctness, perturbation theory, stability.Let A be an operator with nonempty resolvent set p(A) in a Banach space E (hence, A is closed and the domain D(A) is dense in E). For k > 0, we consider the abstract Cauchy problemDefinition 1. A 8olutioa of Eq. (1) is a function u(t) twice strongly differentiable for t > 0, ranging in D(A), and satisfying the equation.Definition 2. Problem (1), (2) is said to be uniformly correct if there exists an operator function Yk(t, A) on E that commutes with A and numbers M > 1 and w > 0 such that for each u0 E D(A) the function Yk(t, A)u0 is the unique solution of (1), (2), and moreover, IIYk(t, A)II < Mexp(~t). (3)The function Yt(t, A) will be referred to as an operator Beasel function; the set of operators for which problem (1), (2) is uniformly correct will be denoted by Gt.If A is the generator of the operator cosine function C(t, A) (see the terminology in [1]), that is, A e Go, then A E Gt for each k > 0 [2]. Note that the embedding Go C Gk is proper (in [3] one can find an example of an operator A(k) E Gt \ Co).In the present paper we study the inclusion A + B E Gk for the perturbed operator under the condition A E Gt. The theory of generators of semigroups and cosine functions can be found, e.g., in [1,4].If Al, A2 E Go, then, in general, the operator A1 + A2 (or its closure) does not belong to Go even if C(t, A1) commutes with C(t, A2) (see [1]). However, the following assertion is valid.

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“…Proposition 8 (see [25]). Problem (12) has a unique solution satisfying condition ( ) if and only if the following Cauchy problem is uniformly well-posed on the space × :…”

Section: Definition 7 One Says That Solution (⋅) Of Problem (12) Satmentioning

confidence: 99%