Dichotomies and asymptotic behaviour for linear differential systems

Author:
James S. Muldowney

Journal:
Trans. Amer. Math. Soc. **283** (1984), 465-484

MSC:
Primary 34D99

DOI:
https://doi.org/10.1090/S0002-9947-1984-0737880-1

MathSciNet review:
737880

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Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions that a system of differential equations have a dichotomy usually require that the matrix be bounded or at least that some restriction be placed on the rate of growth or decay of solutions. Here three sets of necessary and sufficient conditions for a dichotomy which do not impose such a restriction are given in terms of Liapunov functions. Each of the theorems gives practical criteria for a dichotomy including the extension to unbounded matrices of criteria which depend on a concept of diagonal dominance for . An asymptotic analysis is also given for subspaces of the solution set by means of the associated compound equations.

**[1]**Dennis D. Berkey,*Comparative exponential dichotomies and column diagonal dominance*, J. Math. Anal. Appl.**55**(1976), no. 1, 140–149. MR**0417506**, https://doi.org/10.1016/0022-247X(76)90283-3**[2]**W. A. Coppel,*Stability and asymptotic behavior of differential equations*, D. C. Heath and Co., Boston, Mass., 1965. MR**0190463****[3]**W. A. Coppel,*Dichotomies in stability theory*, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR**0481196****[4]**Ju. 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****[5]**Philip Hartman,*The existence of large or small solutions of linear differential equations*, Duke Math. J.**28**(1961), 421–429. MR**0130432****[6]**A. C. Lazer,*Characteristic exponents and diagonally dominant linear differential systems*, J. Math. Anal. Appl.**35**(1971), 215–229. MR**0280818**, https://doi.org/10.1016/0022-247X(71)90246-0**[7]**Jack W. Macki and James S. Muldowney,*The asymptotic behaviour of solutions to linear systems of ordinary differential equations*, Pacific J. Math.**33**(1970), 693–706. MR**0268463****[8]**Marvin Marcus and Henryk Minc,*A survey of matrix theory and matrix inequalities*, Allyn and Bacon, Inc., Boston, Mass., 1964. MR**0162808****[9]**José Luis Massera and Juan Jorge Schäffer,*Linear differential equations and function spaces*, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966. MR**0212324****[10]**James S. Muldowney,*On the dimension of the zero or infinity tending sets for linear differential equations*, Proc. Amer. Math. Soc.**83**(1981), no. 4, 705–709. MR**630041**, https://doi.org/10.1090/S0002-9939-1981-0630041-9**[11]**Kenneth J. Palmer,*A diagonal dominance criterion for exponential dichotomy*, Bull. Austral. Math. Soc.**17**(1977), no. 3, 363–374. MR**0481197**, https://doi.org/10.1017/S0004972700010649**[12]**Binyamin Schwarz,*Totally positive differential systems*, Pacific J. Math.**32**(1970), 203–229. MR**0257466****[13]**E. C. Titchmarsh,*The theory of functions*, Oxford Univ. Press, London, 1960.**[14]**Taro Yoshizawa,*Stability theory by Liapunov’s second method*, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966. MR**0208086**

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DOI:
https://doi.org/10.1090/S0002-9947-1984-0737880-1

Article copyright:
© Copyright 1984
American Mathematical Society