Volume 11, no. 3Pages 85 - 102

Causal Relations in Support of Implicit Evolution Equations

N. Sauer, J. Banasiak, Wha-Suck Lee
This is a brief exposition of dynamic systems approaches that form the basis for linear implicit evolution equations with some indication of interesting applications. Examples in infinite-dimensional dissipative systems and stochastic processes illustrate the fundamental notions underlying the use of double families of evolution equations intertwined by the empathy relation. Kisy'nski's equivalent formulation of the Hille-Yosida theorem highlights the essential differences between semigroup theory and the theory of empathy. The notion of K-bounded semigroups, a more direct approach to implicit equations, and related to empathy in a different way, is included in the survey.
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Keywords
implicit equations; empathy theory; semigroups.
References
1. Hille E., Phillips R.S. Functional Analysis and Semi-Groups, Providence, American Mathematical Society, 1957.
2. Arendt W., Batty C.J.K., Hieber M., Neubrander F. Vector-Valued Laplace Transforms and Cauchy Problems. Basel, Boston, Berlin, Birkh'auser, 2001.
3. Sauer N. Linear Evolution Equations in Two Banach Spaces, Proceedings of the Royal Society of Edinburgh. Series A: Mathematics, 1982, vol. 91, no. 3-4, pp. 287-303.
4. Kisy'nski J. Around Widder's Characterization of the Laplace Transform of an Element of L^{infty}(R^+). Annales Polonici Mathematici, 2000, vol. 74, pp. 161-200. DOI: 10.4064/ap-74-1-161-200
5. Widder D.V. The Laplace Transform. Princeton, Princeton University Press, 1941.
6. Krein S.G. Linear Differential Operators in Banach Space. Providence, American Mathematical Society, 1972.
7. Sauer N. Empathy Theory and the Laplace Transform. Warsaw, Institute of Mathematics Polish Academy of Sciences, 1997, vol. 38, pp. 325-338.
8. Sauer N., Singleton J.E. Evolution Operators Related to Semigroups of Class (A). Semigroup Forum, 1987, vol. 35, pp. 317-335.
9. Sauer N., Van der Merwe A.J. Eigenvalue Problems with the Spectral Parameter also in the Boundary Condition. Quaestiones Mathematicae, 1982, vol. 5, no. 1, pp. 1-27.
10. Showalter R.E. Partial Differential Equations of Sobolev-Galpern Type. Pacific Journal of Mathematics, 1969, vol. 31, no. 3, pp. 787-794.
11. Showalter R.E., Ting T.W. Pseudo-Parabolic Partial Differential Equations. Journal on Mathematical Analysis, 1970, vol. 1, no. 1, pp. 214-231. DOI: 10.1137/0501001
12. Van der Merwe A.J. B-Evolutions and Sobolev Equations. Applicable Analysis, 1988, vol. 29, no. 1-2, pp. 91-105. DOI: 10.1080/00036818808839775
13. Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Marcel Dekker, 1998.
14. Belleni-Morante A. B-Bounded Semigroups and Applications. Annali di Matematica Pura ed Applicata. Serie Quarta, 1996, vol. 170, no. 1, pp. 359-376. DOI: 10.1007/BF01758995
15. Belleni-Morante A., Totaro S. The Successive Reflection Method in Three-Dimensional Particle Transport. Journal of Mathematical Physics, 1996, vol. 37, no. 6, pp. 2815-2823. DOI: 10.1063/1.531541
16. Banasiak J. Generation Results for B-Bounded Semigroups. Annali di Matematica Pura ed Applicata. Serie Quarta, 1998, vol. 175, no. 1, pp. 307-326. DOI: 10.1007/BF01783690
17. Arlotti L. On B-Bounded Semigroups as a Generalization of C_0-Semigroups. Zeitschrift fur Analysis und ihre Anwendungen, 2000, vol. 19, no. 1, pp. 23-34. DOI: 10.4171/ZAA/936
18. Banasiak J. B-Bounded Semigroups and Implicit Evolution Equations. Abstract and Applied Analysis, 2000, vol. 5, no. 1, pp. 13-32 DOI: 10.1155/S1085337500000087
19. Arlotti L. A New Characterization of B-Bounded Semigroups with Application to Implicit Evolution Equations. Abstract and Applied Analysis, 2000, vol. 5, no. 4, pp. 227-243. DOI: 10.1155/S1085337501000331
20. Banasiak J., Singh V. B-Bounded Semigroups and C-Existence Families. Taiwanese Journal of Mathematics, 2002, vol. 6, no. 1, pp. 105-125. DOI: 10.11650/twjm/1500407403
21. Haraux A. Linear Semigroups in Banach Spaces. Harlow, Longman Scientific and Technical, 1986.
22. Banasiak J. B-Bounded Semigroups Existence Families and Implicit Evolution Equations. Basel, Birkhauser, 2000, vol. 42, pp. 25-34.
23. Atay F.M., Roncoroni L. Lumpability of Linear Evolution Equations in Banach Spaces. Evolution Equations and Control Theory, 2017, vol. 6, no. 1, pp. 15-34. DOI: 10.3934/eect.2017002
24. Sauer N. The Friedrichs Extension of a Pair of Operators. Quaestiones Mathematicae, 1989, vol. 12, no. 3, pp. 239-249. DOI: 10.1080/16073606.1989.9632181
25. Grobbelaar-Van Dalsen M., Sauer N. Dynamic Boundary Conditions for the Navier-Stokes Equations. Proceedings of the Royal Society of Edinburgh. Series A: Mathematics, 1989, vol. 113, no. 1-2, pp. 1-11. DOI: 10.1017/S030821050002391X
26. Van der Merwe A.J. Closed Extensions of a Pair of Linear Operators and Dynamic Boundary Value Problems. Applicable Analysis, 1996, vol. 60, no. 1-2, pp. 85-98.
27. Van der Merwe A.J. Perturbations of Evolution Equations. Pretoria, University of Pretoria, 1993.
28. Van der Merwe A.J. Perturbations of Evolution Equations. Applicable Analysis, 1996, vol. 62, no. 3-4, pp. 367-380.
29. Arendt W. Vector-Valued Laplace Transforms and Cauchy Problems. Israel Journal of Mathematics, 1987, vol. 59, no. 3, pp. 327-352.
30. Favini A. Laplace Transform Method for a Class of Degenerate Evolution Problems. Rend. Mat., 1979, vol. 12, pp. 511-536.
31. Brown T.J., Sauer N. Double Families of Integrated Evolution Operators. Journal of Evolution Equations, 2004, vol. 4, no. 4, pp. 567-590.
32. Chojnacki W. On the Equivalence of a Theorem of Kisy'nski and the Hille-Yosida Generation Theorem. Proceedings of the American Mathematical Society, 1998, vol. 126, no. 2, pp. 491-497.
33. Lee W.-S., Sauer N. An Algebraic Approach to Implicit Evolution Equations. Bulletin of the Polish Academy of Sciences Mathematics, 2015, vol. 63, no. 1, pp. 33-40.
34. Lee W.-S., Sauer N. Intertwined Evolution Operators. Semigroup Forum, 2017, vol. 94, pp. 204-228.
35. Lee W.-S., Sauer N. Intertwined Markov Processes: the Extended Chapman-Kolmogorov Equation. Proceedings of the Royal Society of Edinburgh. Series A: Mathematics, 1994, vol. 148A, pp. 123-131.