An Algebraic Approach to Implicit Evolution Equations

• Autorzy: Lee W.-S. , Sauer N.

Identyfikatory

DOI

10.4064/ba63-1-5

• Języki publikacji: EN

Abstrakty

EN

A Banach algebra homomorphism on the convolution algebra of integrable functions is the essence of Kisyński's equivalent formulation of the Hille–Yosida theorem for analytic semigroups. For the study of implicit evolution equations the notion of empathy happens to be more appropriate than that of semigroup. This approach is based upon the intertwining of two families of evolution operators and two families of pseudoresolvents. In this paper we show that the Kisyński approach can be adapted to empathy theory. The adaptation highlights the essential differences between semigroup theory and the theory of empathy.

Słowa kluczowe

EN

evolution operators; implicit evolution equation; Hille-Yosida-Kisyński generation

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 1
• Strony: 33--40
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Lee W.-S.

• Department of Mathematics and Applied Mathematics, University of Pretoria, 0028 Pretoria, South Africa

autor

Sauer N.

• Department of Mathematics and Applied Mathematics, University of Pretoria, 0028 Pretoria, South Africa

Bibliografia

• [1] W. Arendt, C. J. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems: The General Theory, Monogr. Math. 96, Birkhäuser, 2001.
• [2] A. Favini, Laplace transform method for a class of degenerate evolution problems, Rend. Mat. 12 (1979), 511-536.
• [3] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Dekker, New York, 1999.
• [4] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., 2000.
• [5] J. Kisyński, The Widder spaces, representations of the convolution algebra L1(R)+ and one parameter semigroups of operators, Preprint 588, Inst. Math., Polish Acad. Sci., Warszawa, 1998.
• [6] T. W. Palmer, Banach Algebras and The General Theory of *-Algebras. Volume I: Algebras and Banach Algebras, Encyclopedia Math. Appl. 49, Cambridge Univ. Press, 1994.
• [7] N. Sauer, Linear evolution equations in two Banach spaces, Proc. Roy. Soc. Edinburgh Sect. A 91 (1982), 287-303.
• [8] N. Sauer, Empathy theory and the Laplace transform, in: Linear Operators, Banach Center Publ. 38, Inst. Math., Polish Acad. Sci., Warszawa, 1997, 325-338.
• [9] N. Sauer and J. E. Singleton, Evolution operators in empathy with a semigroup, Semigroup Forum 39 (1989), 85-94.
• [10] R. E. Showalter and T. W. Ting, Partial differential equations of Sobolev-Galpern type, Pacific J. Math. 31 (1969), 787-794.
• [11] D. V. Widder, The Laplace Transform, 2nd printing, Princeton Univ. Press, 1946.