Skip to main content
SpringerLink
Log in

New Progress on Operator Semigroups and Linear Systems in Abstract Spaces

  • Chapter
Functional Analysis in China

Part of the book series: Mathematics and Its Applications ((MAIA,volume 356))

  • 352 Accesses

Abstract

Since the fifties, operator semigroups have attracted much attention because they are closely related to Cauchy problems which are very important to applications, and a complete theory on them is established. For the general theory of C 0 semigroups, refer to [1, 2]. Since the seventies, especially since the eighties, Chinese mathematicians have obtained a series of important results in the field of operator semigroups and abstract Cauchy problems. This note briefly outlines the main results in this area accomplished in China during this period. Of course, it can not include all the results obtained. The contents are divided into three parts: stability of C 0 semigroups, integrated semigroups and C-semigroups, and higher order differential equations in abstract spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hille, E. and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ., Vol. 31 Providence R. I., 1957.

    Google Scholar 

  2. Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1983.

    MATH  Google Scholar 

  3. Huang, F. L., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs., 1 (1985), 1, 43–55.

    MATH  Google Scholar 

  4. Gearhart, L., Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385–394.

    Article  MathSciNet  MATH  Google Scholar 

  5. Huang, F. L. and K. S. Liu, On the exponential stability of linear dynamical systems in Hilbert spaces, Kexue Tingbao, (1987), 3, 161–163.

    Google Scholar 

  6. Yao, P. F. and D. X. Feng, Characteristical conditions for exponential stability of C 0 semigroups, Chinese Bulletin of Science, 37 (1993).

    Google Scholar 

  7. Pritchard, A. J. and J. Zabczyk, Stability and stabilizability of infinite dimensional systems, SIAM Review, 23 (1981), 1, 25–52.

    Article  MathSciNet  Google Scholar 

  8. Huang, F. L., Exponential stability of linear systems in Banach spaces, Chin. Ann. of Math., 10B (1989), 3, 332–340.

    Google Scholar 

  9. Feng, D. X. and P. F. Yao, Paley-Wiener theorem and asymptotic behavior of C0 semigroups, accepted for publication in Acta Math. Appl. Sinica, 1995.

    Google Scholar 

  10. Luo, Y. H. and D. X. Feng, On the equi-exponential stability for a family of C 0 semigroups, to appear.

    Google Scholar 

  11. Slemrod, M., Asymptotic behavior of C 0 semigroups as determined by the spectrum of the generator, Indiana Math. J., 25 (1976), 8, 783–792.

    Article  MathSciNet  Google Scholar 

  12. Clément, Ph., H. J. A. M. Heijmans et al., One-Parameter Semigroups, North-Holland, Amsterdam - New York - Oxford - Tokyo, 1987.

    MATH  Google Scholar 

  13. Huang, F. L., Asymptotic stability for linear dynamical systems in Banach spaces, Kexue Tongbao, 10 (1983), 584–586.

    Google Scholar 

  14. Huang, F. L., Spectral properties and stability of one-parameter semigroups, J. Diff. Eqs., 104 (1993), 182–195.

    Article  MATH  Google Scholar 

  15. Huang, F. L., Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Diff. Eqs., 104 (1993), 182–195.

    Article  MATH  Google Scholar 

  16. Huang, F. L., On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control & Optimization, 26 (1988), 3, 714–724.

    Article  MATH  Google Scholar 

  17. Arendt, W. and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837–852.

    Article  MathSciNet  MATH  Google Scholar 

  18. Da Prato, G., Semigruppi regolarizzabili, Ricerche di Matematica, 15 (1966), 2, 223–248.

    MathSciNet  MATH  Google Scholar 

  19. Davies, E. B. and M. M. H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc., 55 (1987), 1, 181–208.

    Article  MathSciNet  MATH  Google Scholar 

  20. Arendt, W., Proc. London Math. Soc., 54 (1987), 321–349.

    Article  MathSciNet  MATH  Google Scholar 

  21. Zheng, Q. and Y. Lei, Exponentially bounded C-semigroups and integrated semigroups with nondensely defined generators, I: Approximation; II: Perturbation; III: Analyticity, Acta Math. Sci., 13 (1993), 3; 13 (1993), 4; 14 (1994), 1.

    Google Scholar 

  22. Zheng, Q. and Y. Lei, Adjoint semigroups of exponentially bounded C-semigroups (in Chinese), Math. Appl., 6 (1993), Special Issue.

    Google Scholar 

  23. Zheng, Q., Perturbation and approximation of integrated semigroups, Acta Math. Sinica (new series), 9 (1993), 3.

    Google Scholar 

  24. Zheng, Q. and Y. Lei, On the integrated C-semigroups (in Chinese), J. Huazhong Univ. Sci. Tech., 20 (1992), 5, 181–187.

    Google Scholar 

  25. Zheng, Q., Integrated cosine functions, to appear.

    Google Scholar 

  26. Zheng, Q., Higher order abstract Cauchy problems with n-closeness, Math. Japan, 38 (1993), 531–539.

    MATH  Google Scholar 

  27. Zheng, Q., Some operator matrices as integrated semigroup generators (in Chinese), Acta Math. Sinica, 36 (1993), 456–467.

    MathSciNet  MATH  Google Scholar 

  28. Zheng, Q., Some operator matrices as integrated semigroup operators (in Chinese), Acta Math. Sinica, 36 (1993), 456–467.

    MathSciNet  MATH  Google Scholar 

  29. Zheng, Q., Y. Lei and W. Yi, Semigroups of operators and polynomials of generators of bounded strongly continuous groups, Proc. London Math. Soc., to appear.

    Google Scholar 

  30. Zheng, Q. and Y. Lei, C-semigroups, integrated semigroups and integrated solutions of abstract Cauchy problem, in: Applied Functional Analysis Ser., Vol. 1, ed. M. Z. Yang, World Publishing Corporation, Beijing, 1993.

    Google Scholar 

  31. Fattorini, H. O., Extension and behavior at infinity of solutions of certain linear operational differential equations, Pacific J. Math., 33 (1970), 583–615.

    MathSciNet  Google Scholar 

  32. Xiao, T. J., The well posedness of a class of complete second order differential equations in Banach spaces, J. Sichuan Univ. Nat. Sci. Edi., 25 (1988), 421–425.

    MATH  Google Scholar 

  33. Neubrander, F., Integrated semigroups and their applications to complete second problems, Semigroup Forum, 38 (1989), 233–251.

    Article  MathSciNet  MATH  Google Scholar 

  34. Xiao, T. J. and J. Liang, On complete second order differential linear equations in Banach spaces, Pacific J. Math., 142 (1990), 175–197.

    MathSciNet  Google Scholar 

  35. Xiao, T. J., The well posedness of Cauchy problem of complete second order differential equations in Fréchet spaces, Acta Mathematica Sinica, 35 (1992), 354–363.

    MathSciNet  MATH  Google Scholar 

  36. Xiao, T. J., Second order linear differential equations with almost periodic solutions, Acta Mathematica Sinica, New Series, 7 (1991), 354–359.

    MATH  Google Scholar 

  37. Xiao, T. J., J. Yunnan Norm. Nat. Sci. Edi., 12 (1992), 4, 10–13.

    Google Scholar 

  38. Xiao, T. J., The Cauchy problems of higher order abstract differential equations, Chin. Ann. of Math., 14A (1993), 5, 23–35.

    Google Scholar 

  39. Xiao, T. J., Complete second order linear differential equations with almost periodic solutions, J. Math. Anal. Appl., 163 (1992), 136–146.

    Article  MathSciNet  Google Scholar 

  40. Xiao, T. J., On some worthy problems in abstract differential equations, J. Yunnan Norm. Univ. Nat. Sci. Edi., 11 (1991), 4, 9–15.

    Google Scholar 

  41. Sova, M., Ann. Scuola Norm. Sup. Pisa, 22 (1968), 67–100.

    MathSciNet  MATH  Google Scholar 

  42. Xiao, T. J., A note on the propagators of second order linear differential equations in Hilbert spaces, Proc. Amer. Math. Soc., 113 (1991), 663–667.

    Article  MathSciNet  Google Scholar 

  43. Xiao, T. J., Complete second order linear differential equations, Kexue Tongbao, 33 (1988), 1274–1275.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Systems Science, Academia Sinica, Beijing, China

    Dexing Feng

Authors
  1. Dexing Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Academia Sinica, Beijing, China

    Bingren Li

  2. Department of Mathematics, Nanjing University, Nanjing, China

    Shengwang Wang

  3. Department of Mathematics, Fudan University, Shanghai, China

    Shaozong Yan

  4. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Chung-Chun Yang

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Kluwer Academic Publishers

About this chapter

Cite this chapter

Feng, D. (1996). New Progress on Operator Semigroups and Linear Systems in Abstract Spaces. In: Li, B., Wang, S., Yan, S., Yang, CC. (eds) Functional Analysis in China. Mathematics and Its Applications, vol 356. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0185-8_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-0185-8_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6567-2

  • Online ISBN: 978-94-009-0185-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation