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On Maximal Regularity and Semivariation of Cosine Operator Functions
Author(s) -
Chyan D.K.,
Shaw S.Y.,
Piskarev S.
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799007073
Subject(s) - trigonometric functions , bounded function , operator (biology) , mathematics , differentiable function , generator (circuit theory) , bounded operator , function (biology) , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , physics , geometry , power (physics) , biochemistry , chemistry , repressor , quantum mechanics , evolutionary biology , biology , transcription factor , gene
It is proved that a cosine operator function C (·), with generator A , is locally of bounded semivariation if and only if u ″( t ) = Au( t )+ f ( t ), t >0, u (0), u ′(0)∈ D ( A ), has a strong solution for every continuous function f , if and only if the function∫ 0 t∫ 0 t ‐ 8C ( τ ) f ( s ) d τ d s , t > 0 , is twice continuously differentiable for every continuous function f , that is, C (·) has the maximal regularity property if and only if A is a bounded operator. Some other characterisations of bounded generators of cosine operator functions are also established in terms of their local semivariations.