α-Times Integrated C-Semigroups

ABSTRACT

The α-times integrated C semigroups, α > 0, are introduced and analyzed. The Laplace inverse transformation for α-times integrated C semigroups is obtained, some known results are generalized.

The α-times integrated C semigroups, α > 0, are introduced and analyzed. The Laplace inverse transformation for α-times integrated C semigroups is obtained, some known results are generalized.

Cite this paper

M. Liu, D. Liao, Q. Zhu and F. Wang, "α-Times Integrated C-Semigroups,"*Advances in Pure Mathematics*, Vol. 2 No. 3, 2012, pp. 211-215. doi: 10.4236/apm.2012.23030.

M. Liu, D. Liao, Q. Zhu and F. Wang, "α-Times Integrated C-Semigroups,"

References

[1] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, New York, 1983. doi:10.1007/978-1-4612-5561-1

[2] Q. Zhao, G. T. Zhu and D. X. Feng, “Generalized Operator Semigroup and Well-Posedness of Singular Distributed Parameter Systems,” Science in China, Series A, Vol. 40, No. 5, 2010, pp. 477-495.

[3] T.-J. Xiao and J. Liang, “Approximation of Laplace Trans- forms and Integrated Semigroups,” Journal of Functional Analysis, Vol. 172, No. 1, 2000, pp. 202-220. doi:10.1006/jfan.1999.3545

[4] X. Q. Song, “Spectral Map-ping Theorems for C-Semi-groups,” Journal of Mathematical Research & Exposition, Vol. 40, No. 5, 1996, pp. 526-530.

[5] Q. Zheng, “Pertubations and Approximations of Integrate Semigroups,” Acta Mathematica Sinica, New Series, Vol. 9, No. 3, 1993, pp. 252-260. doi:10.1007/BF02582903

[6] Q. Zheng, “Integral Semigroup and Abstract Cauchy Problem,” Advances in Mathematics, Vol. 21, No. 3, 1992, pp. 257-273.

[7] K. L. Lang and G. J. Yang, “Local c-Semigroup and Abstract Cauchy Problem,” Applied Mathematics, Vol. 11, No. 4, 1998, pp. 35-37.

[8] I. Miyadera and N. Tanaka, “A Remark on Exponentially Bounded C-Semigroups,” Proceedings of the Japan Aca- demy Series A, Vol. 66, No. 2, 1990, pp. 31-34.

[9] W. Arendt, “Vec-tor-Valued Laplace Transforms and Cauchy Problems,” Israel Journal of Mathematics, Vol. 59, No. 3, 1987, pp. 327-352. doi:10.1007/BF02774144

[10] W. Arendt, “Resolvent Positive Operators,” Proceedings London Mathematical Society, Vol. 54, No. 2, 1978, pp. 321-349. doi:10.1112/plms/s3-54.2.321

[11] R. Delaubenfels, “Existence and Uniqueness Families for the Abstract Cauchy Problem,” London Mathematical Society, Vol. 44, No. 2, 1991, pp. 310-322.

[12] S. W. Wang, “Mild Integrated C-Existence Families,” Studia Mathematics, Vol. 112, No. 3, 1995, pp. 251-266.

[13] M. C. Gao, “Mild Integrated C-Existence Families and Abstract Cauchy Problem,” Northeast Mathematics, Vol. 14, No. 1, 1998, pp. 95-102.

[14] N. Tanaka and I. Miyadera, “Exponentially Bounded C Semigroups and Integrated Semigroups,” Tokyo Journal of Mathematics, Vol. 12, No. 3, 1989, pp. 99-115. doi:10.3836/tjm/1270133551

[15] W. Arendt, “Vector Valued Laplace Tranforms and Cauchy Problems,” Israel Journal of Mathematics, Vol. 59, No. 3, 1987, pp. 327-352. doi:10.1007/BF02774144

[16] M. Mijatovic and S. pilipovic, “α-Times Integrated Semi- group,” Journal of Mathematical Analysis and Applications, Vol. 210, No. 2, 1997, pp. 790-803. doi:10.1006/jmaa.1997.5436

[17] Existence Families, “Functional calculi and Evolution Equations,” Lecture Notes in Mathematics, Springer-Verlag, New York, 1994.

[18] M. Hieber, “Laplace Transforms and α-Times Integrated Semigroup,” Forum Mathematicum, Vol. 3, No. 3, 1991, pp. 595-612. doi:10.1515/form.1991.3.595

[1] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, New York, 1983. doi:10.1007/978-1-4612-5561-1

[2] Q. Zhao, G. T. Zhu and D. X. Feng, “Generalized Operator Semigroup and Well-Posedness of Singular Distributed Parameter Systems,” Science in China, Series A, Vol. 40, No. 5, 2010, pp. 477-495.

[3] T.-J. Xiao and J. Liang, “Approximation of Laplace Trans- forms and Integrated Semigroups,” Journal of Functional Analysis, Vol. 172, No. 1, 2000, pp. 202-220. doi:10.1006/jfan.1999.3545

[4] X. Q. Song, “Spectral Map-ping Theorems for C-Semi-groups,” Journal of Mathematical Research & Exposition, Vol. 40, No. 5, 1996, pp. 526-530.

[5] Q. Zheng, “Pertubations and Approximations of Integrate Semigroups,” Acta Mathematica Sinica, New Series, Vol. 9, No. 3, 1993, pp. 252-260. doi:10.1007/BF02582903

[6] Q. Zheng, “Integral Semigroup and Abstract Cauchy Problem,” Advances in Mathematics, Vol. 21, No. 3, 1992, pp. 257-273.

[7] K. L. Lang and G. J. Yang, “Local c-Semigroup and Abstract Cauchy Problem,” Applied Mathematics, Vol. 11, No. 4, 1998, pp. 35-37.

[8] I. Miyadera and N. Tanaka, “A Remark on Exponentially Bounded C-Semigroups,” Proceedings of the Japan Aca- demy Series A, Vol. 66, No. 2, 1990, pp. 31-34.

[9] W. Arendt, “Vec-tor-Valued Laplace Transforms and Cauchy Problems,” Israel Journal of Mathematics, Vol. 59, No. 3, 1987, pp. 327-352. doi:10.1007/BF02774144

[10] W. Arendt, “Resolvent Positive Operators,” Proceedings London Mathematical Society, Vol. 54, No. 2, 1978, pp. 321-349. doi:10.1112/plms/s3-54.2.321

[11] R. Delaubenfels, “Existence and Uniqueness Families for the Abstract Cauchy Problem,” London Mathematical Society, Vol. 44, No. 2, 1991, pp. 310-322.

[12] S. W. Wang, “Mild Integrated C-Existence Families,” Studia Mathematics, Vol. 112, No. 3, 1995, pp. 251-266.

[13] M. C. Gao, “Mild Integrated C-Existence Families and Abstract Cauchy Problem,” Northeast Mathematics, Vol. 14, No. 1, 1998, pp. 95-102.

[14] N. Tanaka and I. Miyadera, “Exponentially Bounded C Semigroups and Integrated Semigroups,” Tokyo Journal of Mathematics, Vol. 12, No. 3, 1989, pp. 99-115. doi:10.3836/tjm/1270133551

[15] W. Arendt, “Vector Valued Laplace Tranforms and Cauchy Problems,” Israel Journal of Mathematics, Vol. 59, No. 3, 1987, pp. 327-352. doi:10.1007/BF02774144

[16] M. Mijatovic and S. pilipovic, “α-Times Integrated Semi- group,” Journal of Mathematical Analysis and Applications, Vol. 210, No. 2, 1997, pp. 790-803. doi:10.1006/jmaa.1997.5436

[17] Existence Families, “Functional calculi and Evolution Equations,” Lecture Notes in Mathematics, Springer-Verlag, New York, 1994.

[18] M. Hieber, “Laplace Transforms and α-Times Integrated Semigroup,” Forum Mathematicum, Vol. 3, No. 3, 1991, pp. 595-612. doi:10.1515/form.1991.3.595