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## Studia Mathematica

2000 | 142 | 3 | 201-217
Tytuł artykułu

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

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Warianty tytułu
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EN
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EN
This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u'(t) = Au(t) + \frac{t^{α-1}}{Γ(α)}x$, 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 $(λ-A)^{-1}C(X)$.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
201-217
Opis fizyczny
Daty
wydano
2000
otrzymano
1998-02-17
poprawiono
2000-04-03
Twórcy
autor
• Department of Mathematics, Fu-Jen University, Hsin-Chuang, Taipei, Taiwan
autor
• Department of Mathematics, National Central University, Chung-Li, Taiwan
Bibliografia
• [1] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352.
• [2] I. Cioranescu and G. Lumer, On K(t)-convoluted semigroups, in: Pitman Res. Notes Math. Ser. 324, Glasgow, 1994, 86-93.
• [3] E. B. Davies and M. M. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc. 55 (1987), 181-208.
• [4] R. deLaubenfels, C-semigroups and the Cauchy problem, J. Funct. Anal. 111 (1993), 44-61.
• [5] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, 1985.
• [6] M. Hieber, Laplace transforms and α-times integrated semigroups, Forum Math. 3 (1991), 595-612.
• [7] C.-C. Kuo and S.-Y. Shaw, On strong and weak solutions of abstract Cauchy problems, preprint.
• [8] Y.-C. Li, Integrated C-semigroups and C-cosine functions of operators on locally convex spaces, Ph.D. dissertation, National Central Univ., 1991.
• [9] Y.-C. Li and S.-Y. Shaw, N-times integrated C-semigroups and the abstract Cauchy problem, Taiwanese J. Math. 1 (1997), 75-102.
• [10] Y.-C. Li and S.-Y. Shaw, On generators of integrated C-semigroups and C-cosine functions, Semigroup Forum 47 (1993), 29-35.
• [11] I. Miyadera, On the generators of exponentially bounded C-semigroups, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 239-242.
• [12] I. Miyadera, M. Okubo and N. Tanaka, On integrated semigroups which are not exponentially bounded, ibid. 69 (1993), 199-204.
• [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.
• [14] F. Neubrander, Integrated semigroups and their applications to the abstract Cauchy problem, Pacific J. Math. 135 (1988), 111-155.
• [15] N. Tanaka and I. Miyadera, Exponentially bounded C-semigroups and integrated semigroups, Tokyo J. Math. 12 (1989), 99-115.
• [16] N. Tanaka and I. Miyadera, C-semigroups and the abstract Cauchy problem, J. Math. Anal. Appl. 170 (1992), 196-206.
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Bibliografia
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