## Taiwanese Journal of Mathematics

### ON PERTURBATION OF $\alpha$-TIMES INTEGRATED $C$-SEMIGROUPS

Chung-Cheng Kuo

#### Abstract

Let $\alpha \geq 0$, and $C$ be a bounded linear injection on a complex Banach space $X$. We first show that if $A$ generates an exponentially bounded nondegenerate $\alpha$-times integrated $C$-semigroup $S_{\alpha}(\cdot)$ on $X$, $B$ is a bounded linear operator on $\overline{D(A)}$ such that $BC = CB$ on $\overline{D(A)}$ and $BA \subset AB$, then $A+B$ generates an exponentially bounded nondegenerate $\alpha$-times integrated $C$-semigroup $T_{\alpha}(\cdot)$ on $X$. Moreover, $T_{\alpha}(\cdot)$ is also exponentially Lipschitz continuous or norm continuous if $S_{\alpha}(\cdot)$ is. We show that the exponential boundedness of $T_{\alpha}(\cdot)$ can be deleted and $\alpha$-times integrated $C$-semigroups can be extended to the context of local $\alpha$-times integrated $C$-semigroups when $R(C) \subset \overline{D(A)}$ and $BS_{\alpha}(\cdot) = S_{\alpha}(\cdot)B$ on $\overline{D(A)}$ both are added. Moreover, $T_{\alpha}(\cdot)$ is also locally Lipschitz continuous or norm continuous if $S_{\alpha}(\cdot)$ is. We show that $A+B$ generates a nondegenerate local $\alpha$-times integrated $C$-semigroup $T_{\alpha}(\cdot)$ on $X$ if $A$ generates a nondegenerate local $\alpha$-times integrated $C$-semigroup $S_{\alpha}(\cdot)$ on $X$ and $B$ is a bounded linear operator on $X$ such that either $BC = CB$, $BS_{\alpha} = S_{\alpha}B$ on $X$; or $BC = CB$ on $\overline{D(A)}$ and $BA \subset AB$. Moreover, $T_{\alpha}(\cdot)$ is also locally Lipschitz continuous, (norm continuous, exponentially bounded or exponentially Lipschitz continuous) if $S_{\alpha}(\cdot)$ is.

#### Article information

Source
Taiwanese J. Math., Volume 14, Number 5 (2010), 1979-1992.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406027

Digital Object Identifier
doi:10.11650/twjm/1500406027

Mathematical Reviews number (MathSciNet)
MR2724144

Zentralblatt MATH identifier
1239.47037

#### Citation

Kuo, Chung-Cheng. ON PERTURBATION OF $\alpha$-TIMES INTEGRATED $C$-SEMIGROUPS. Taiwanese J. Math. 14 (2010), no. 5, 1979--1992. doi:10.11650/twjm/1500406027. https://projecteuclid.org/euclid.twjm/1500406027

#### References

• [1.] W. Arendt, C. J. K. Batty, H. Hieber and F. Neubrander, Vector-Valued Laplace Transformes and Cauchy Problem, Birkhauser Verlag, Basel-Boston-Berlin, 2001, p. 96.
• [2.] R. DeLaubenfuls, Integrated Semigroups, $\tC$-Semigroups and the Abstract Cauchy Problem, Semigroup Forum, 41 (1990), 83-95.
• [3.] M. Hieber, Integrated Semigroups and Differential Operators on $\tL^p$ Spaces, Math. Ann., 291 (1991), 1-16.
• [4.] M. Hieber, Laplace Transforms and $\alpha$-Times Integrated Semigroups, Forum Math., 3 (1991), 595-612.
• [5.] H. Kellerman and M. Hieber, Integrated Semigroups, J. Funct. Anal., 84 (1989), 160-180.
• [6.] C.-C. Kuo and S.-Y. Shaw, On $\alpha$-Times Integrated $\tC$-Semigroups and the Abstract Cauchy Problem, Studia Math., 142 (2000), 201-217.
• [7.] C.-C. Kuo and S.-Y. Shaw, On Strong and Weak Solutions of Abstract Cauchy Problem, J. Concrete and Applicable Math., 2 (2004), 191-212.
• [8.] M. Li and Q. Zheng, $\alpha$-Times Integrated Semigroups: Local and Global, Studia Math., 154 (2003), 243-252.
• [9.] Y.-C. Li and S.-Y. Shaw, N-Times Integrated $\tC$-Semigroups and the Abstract Cauchy Problem, Taiwanese J. Math., 1 (1997), 75-102.
• [10.] Y.-C. Li and S.-Y. Shaw, Perturbation of Non-exponentially-Bounded $\alpha$-Times Integrated $\tC$-Semigroups, J. Math. Soc. Japan, 55 (2003), 1115-1136.
• [11.] Y.-C. Li and S.-Y. Shaw, On Local $\alpha$-Times Integrated C-Semigroups, Abstract and Applied Anal. Vol. 2007, Article ID34890, 18 pages.
• [12.] M. Mijatović and S. Pilipović, $\alpha$-Times Integrated Semigroups ($\alpha\in\Bbb R^+$), J. Math. Anal. Appl., 210 (1997), 790-803.
• [13.] I. Miyadera, M. Okubo and N. Tanaka, On Integrated Semigroups where are not Exponentially Bounded, Proc. Japan Acad., 69 (1993), 199-204.
• [14.] F. Neubrander, Integrated Semigroups and their Applications to the Abstract Cauchy Problem, Pacific J. Math., 135 (1988), 111-155.
• [15.] F. Neubrander, Integrated Semigroups and their Applications to Complete Second Order Cauchy Problems, Semigroup Forum, 38 (1989), 233-251.
• [16.] S. Nicasie, The Hille-Yosida and Trotter-Kato, Theorems for Integrated Semigroups, J. Math. Anal. Appl., 180 (1993), 303-316.
• [17.] S.-Y. Shaw and C.-C. Kuo, Generation of Local $\tC$-Semigroups and Solvability of the Abstract Cauchy Problems, Taiwanese J. Math., 9 (2005), 291-311.
• [18.] N. Tanaka, On Perturbation Theorey for Exponentially Bounded $\tC$-Semigroups, Semigroup Forum, 41 (1990), 215-236.
• [19.] T.-J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, Lectures Notes in Math. 1701, Springer, 1998.
• [20.] T.-J. Xiao and J. Liang, Approximations of Laplace Transforms and Integrated Semigroups, J. Funct. Anal., 172 (2000), 202-220.
• [21.] Q. Zheng, Perturbations and Approximations of Integrated Semigroups, Acta Math. Sci., 9 (1993), 252-260.