On α-times integrated C-semigroups and the abstract Cauchy problem

Chung-Cheng Kuo; Sen-Yen Shaw

Studia Mathematica (2000)

  • Volume: 142, Issue: 3, page 201-217
  • ISSN: 0039-3223

Abstract

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This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: u ' ( t ) = A u ( t ) + t α - 1 Γ ( α ) x , t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x in ( λ - A ) - 1 C ( X ) .

How to cite

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Kuo, Chung-Cheng, and Shaw, Sen-Yen. "On α-times integrated C-semigroups and the abstract Cauchy problem." Studia Mathematica 142.3 (2000): 201-217. <http://eudml.org/doc/216798>.

@article{Kuo2000,
abstract = {This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u^\{\prime \}(t) = Au(t) + \frac\{t^\{α-1\}\}\{Γ(α)\}x$, t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x in $(λ-A)^\{-1\}C(X)$.},
author = {Kuo, Chung-Cheng, Shaw, Sen-Yen},
journal = {Studia Mathematica},
keywords = {generator; abstract Cauchy problem; α-times integrated C-semigroup; -times integrated -semigroup; Laplace transforms},
language = {eng},
number = {3},
pages = {201-217},
title = {On α-times integrated C-semigroups and the abstract Cauchy problem},
url = {http://eudml.org/doc/216798},
volume = {142},
year = {2000},
}

TY - JOUR
AU - Kuo, Chung-Cheng
AU - Shaw, Sen-Yen
TI - On α-times integrated C-semigroups and the abstract Cauchy problem
JO - Studia Mathematica
PY - 2000
VL - 142
IS - 3
SP - 201
EP - 217
AB - This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u^{\prime }(t) = Au(t) + \frac{t^{α-1}}{Γ(α)}x$, t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x in $(λ-A)^{-1}C(X)$.
LA - eng
KW - generator; abstract Cauchy problem; α-times integrated C-semigroup; -times integrated -semigroup; Laplace transforms
UR - http://eudml.org/doc/216798
ER -

References

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  12. [12] I. Miyadera, M. Okubo and N. Tanaka, On integrated semigroups which are not exponentially bounded, ibid. 69 (1993), 199-204. Zbl0812.47041
  13. [13] I. Miyadera, M. Okubo and N. Tanaka, α-integrated semigroups and abstract Cauchy problems, Mem. School Sci. Engrg. Waseda Univ. 57 (1993), 267-289. 
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  16. [16] N. Tanaka and I. Miyadera, C-semigroups and the abstract Cauchy problem, J. Math. Anal. Appl. 170 (1992), 196-206. Zbl0812.47044

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