Empathy theory and the Laplace transform

Niko Sauer

Banach Center Publications (1997)

  • Volume: 38, Issue: 1, page 325-338
  • ISSN: 0137-6934

Abstract

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This paper is concerned with double families of evolution operators employed in the study of dynamical systems in which cause and effect are represented in different Banach spaces. The main tool is the Laplace transform of vector-valued functions. It is used to define the generator of the double family which is a pair of unbounded linear operators and relates to implicit evolution equations in a direct manner. The characterization of generators for a special class of evolutions is presented.

How to cite

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Sauer, Niko. "Empathy theory and the Laplace transform." Banach Center Publications 38.1 (1997): 325-338. <http://eudml.org/doc/208639>.

@article{Sauer1997,
abstract = {This paper is concerned with double families of evolution operators employed in the study of dynamical systems in which cause and effect are represented in different Banach spaces. The main tool is the Laplace transform of vector-valued functions. It is used to define the generator of the double family which is a pair of unbounded linear operators and relates to implicit evolution equations in a direct manner. The characterization of generators for a special class of evolutions is presented.},
author = {Sauer, Niko},
journal = {Banach Center Publications},
keywords = {double families of evolution operators; dynamical systems; Laplace transform of vector-valued functions; generator; pair of unbounded linear operators; implicit evolution equations},
language = {eng},
number = {1},
pages = {325-338},
title = {Empathy theory and the Laplace transform},
url = {http://eudml.org/doc/208639},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Sauer, Niko
TI - Empathy theory and the Laplace transform
JO - Banach Center Publications
PY - 1997
VL - 38
IS - 1
SP - 325
EP - 338
AB - This paper is concerned with double families of evolution operators employed in the study of dynamical systems in which cause and effect are represented in different Banach spaces. The main tool is the Laplace transform of vector-valued functions. It is used to define the generator of the double family which is a pair of unbounded linear operators and relates to implicit evolution equations in a direct manner. The characterization of generators for a special class of evolutions is presented.
LA - eng
KW - double families of evolution operators; dynamical systems; Laplace transform of vector-valued functions; generator; pair of unbounded linear operators; implicit evolution equations
UR - http://eudml.org/doc/208639
ER -

References

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  3. [CaSh76] R. W. Carroll and R. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York, 1976. 
  4. [CS94] W. L. Conradie and N. Sauer, Empathy, C-semigroups and integrated semigroups, in: Evolution Equations, Proc. Conf. Baton Rouge 1993, G. Ferreyra, G. R. Goldstein and F. Neubrander (eds.), Lecture Notes in Pure and Appl. Math. 168, Marcel Dekker, New York, 1995, 123-132. 
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  7. [DeL91] R. deLaubenfels, Existence and uniqueness families, for the abstract Cauchy problem, J. London Math. Soc. 44 (1991), 310-338. Zbl0766.47011
  8. [DeL94] R. deLaubenfels, Existence Families, Functional Calculus and Evolution Equations, Lecture Notes in Math. 1570, Springer, Berlin, 1994. 
  9. [DU77] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977. 
  10. [Fav79] A. Favini, Laplace transform method for a class of degenerate evolution equations, Rend. Mat. 3-4 (1979), 511-536. 
  11. [Fri58] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333-418. Zbl0083.31802
  12. [HP57] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, 1957. 
  13. [Miy51] I. Miyadera, One parameter semi-groups of operators, J. Math. Tokyo 8 (1951), 23-26. 
  14. [Sau82] N. Sauer, Linear evolution equations in two Banach spaces, Proc. Roy. Soc. Edinburgh 91A (1982), 387-303. 
  15. [Sau95] N. Sauer, Implicit evolution equations and empathy theory, in: Recent Developments in Evolution Equations, Pitman Res. Notes in Math. Ser. 324, A. C. McBride and G. F. Roach (eds.), Longman, Harlow, 1995, 32-39. Zbl0835.47031
  16. [SS87] N. Sauer and J. E. Singleton, Evolution operators related to semigroups of class (A), Semigroup Forum 35 (1987), 317-335. Zbl0617.47030
  17. [SS89] N. Sauer and J. E. Singleton, Evolution operators in empathy with a semigroup, ibid. 39 (1989), 85-94. Zbl0683.47024
  18. [Wid46] D. V. Widder, The Laplace Transform, Princeton Univ. Press, 2nd printing, 1946. 

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