On the proof of characterizations
of the exponential dichotomy
Nguyen Van Minh
Proc. Amer. Math. Soc. 127 (1999), 779-782
Primary 34G10, 47D06; Secondary 47H20
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Abstract: We describe explicitly the generator of the evolutionary semigroup associated with the evolutionary operator generated by the linear differential equation . From this we give a short proof of some known characterizations of the exponential dichotomy of the above mentioned equation.
Aulbach B., Nguyen Van Minh, Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous evolution equations, Abstract and Applied Analysis 1 (1996), 351-380.
L. Dalec′kiĭ and M.
G. Kreĭn, Stability of solutions of differential equations
in Banach space, American Mathematical Society, Providence, R.I.,
1974. Translated from the Russian by S. Smith; Translations of Mathematical
Monographs, Vol. 43. MR 0352639
J. Ciach, R.
Jajte, and A.
Paszkiewicz, On the almost sure approximation of self-adjoint
operators in 𝐿₂(0,1), Math. Proc. Cambridge Philos.
Soc. 119 (1996), no. 3, 537–543. MR 1357063
M. Levitan and V.
V. Zhikov, Almost periodic functions and differential
equations, Cambridge University Press, Cambridge-New York, 1982.
Translated from the Russian by L. W. Longdon. MR 690064
Văn Minh, Semigroups and stability of
nonautonomous differential equations in Banach spaces, Trans. Amer. Math. Soc. 345 (1994), no. 1, 223–241. MR 1242785
Pazy, Semigroups of linear operators and applications to partial
differential equations, Applied Mathematical Sciences, vol. 44,
Springer-Verlag, New York, 1983. MR 710486
Rau R., Hyperbolic evolutionary semigroups on vector valued function spaces, Semigroup Forum 48(1994), 107-118.
Zhikov V.V., On the theory of admissibility of pairs of function spaces, Soviet Math. Dokl. vol. 13(1972), N.4, 1108-1111.
- Aulbach B., Nguyen Van Minh, Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous evolution equations, Abstract and Applied Analysis 1 (1996), 351-380.
- Daleckii Ju. L., Krein M.G., ``Stability of Solutions of Differential Equations in Banach Space'', Amer. Math. Soc., Providence RI, 1974. MR 50:5126
- Latushkin Ju., Montgomery-Smith S., Evolutionary semigroups and Lyapunov theorems in Banach spaces J. Func. Anal. 127(1995), 173-197. MR 96k:47022
- Levitan B.M., Zhikov V.V., ``Almost Periodic Functions and Differential Equations'', Moscow Univ. Publ. House 1978. English translation by Cambridge University Press 1982. MR 84g:34004
- Nguyen Van Minh, Semigroups and stability of nonautonomous differential equations in Banach spaces, Trans. Amer. Math. Soc., 345(1994), N.1, 223-242. MR 95a:34091
- Pazy A., ``Semigroups of Linear Operators and Applications to Partial Differential Equations", Applied Math. Sci. 44, Springer-Verlag, Berlin-New York 1983. MR 85g:47061
- Rau R., Hyperbolic evolutionary semigroups on vector valued function spaces, Semigroup Forum 48(1994), 107-118.
- Zhikov V.V., On the theory of admissibility of pairs of function spaces, Soviet Math. Dokl. vol. 13(1972), N.4, 1108-1111.
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Nguyen Van Minh
Department of Mathematics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan
Received by editor(s):
December 20, 1996
Received by editor(s) in revised form:
June 20, 1997
This note was written during the author’s visit to the Institute of Mathematics, University of Tübingen. The support of the Alexander von Humboldt Foundation is gratefully acknowledged. He also thanks R. Nagel and the Division of Functional Analysis at the Institute for their warm hospitality and constant encouragement. Finally, he thanks the referee for the valuable suggestion to improve the presentation of this paper.
Palle E. T. Jorgensen
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