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

Studia Mathematica (2000)

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

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topKuo, 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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