Abstract
The unstable properties of the linear nonautonomous delay system x′(t) = A(t)x(t) + B(t)x(t − r(t)), with nonconstant delay r(t), are studied. It is assumed that the linear system y′(t) = (A(t) + B(t))y(t) is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay r(t) is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function r(t) and the results depending on the asymptotic properties of the delay function.
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References
E. A. Coddington and N. Levinson: Theory of Ordinary Differential Equations. McGill-Hill, New York, 1975.
K. L. Cooke: Functional differential equations close to differential equations. Bull. Amer. Math. Soc. 72 (1966), 285–288.
W. A. Coppel: On the stability of ordinary differential equations. J. London Math. Soc. 39 (1964), 255–260.
W. A. Coppel: Dichotomies in Stability Theory. Lecture Notes in Mathematics Vol. 629. Springer Verlag, Berlin, 1978.
L. E. Elsgoltz and S. B. Norkin: Introduction to the Theory of Differential Equations with Deviating Arguments. Nauka, Moscow, 1971. (In Russian.)
J. Gallardo and M. Pinto: Asymptotic integration of nonautonomous delay-differential systems. J. Math. Anal. Appl. 199 (1996), 654–675.
K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Populations Dynamics. Kluwer, Dordrecht, 1992.
I. Győri and M. Pituk: Stability criteria for linear delay differential equations. Differential Integral Equations 10 (1997), 841–852.
N. Rouche, P. Habets and M. Laloy: Stability Theory by Liapounov's Second Method. App. Math. Sciences 22. Springer, Berlin, 1977.
J. K. Hale: Theory of Functional Differential Equations. Springer-Verlag, New York, 1977.
J. K. Hale and S. M. Verduyn Lunel: Introduction to Functional Differential Equations. Springer-Verlag, New York, 1993.
R. Naulin: Instability of nonautonomous differential systems. Differential Equations Dynam. Systems 6 (1998), 363–376.
R. Naulin: Weak dichotomies and asymptotic integration of nonlinear differential systems. Nonlinear Studies 5 (1998), 201–218.
R. Naulin: Functional analytic characterization of a class of dichotomies. Unpublished work (1999).
R. Naulin and M. Pinto: Roughness of (h; k)-dichotomies. J. Differential Equations 118 (1995), 20–35.
R. Naulin and M. Pinto: Admissible perturbations of exponential dichotomy roughness. J. Nonlinear Anal. TMA 31 (1998), 559–571.
R. Naulin and M. Pinto: Projections for dichotomies in linear differential equations. Appl. Anal. 69 (1998), 239–255.
M. Pinto: Non autonomous semilinear differential systems: Asymptotic behavior and stable manifolds. Preprint (1997).
M. Pinto: Dichotomy and asymptotic integration. Contributions USACH (1992), 13–22.
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Naulin, R. On the Instability of Linear Nonautonomous Delay Systems. Czechoslovak Mathematical Journal 53, 497–514 (2003). https://doi.org/10.1023/B:CMAJ.0000024498.19392.a9
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DOI: https://doi.org/10.1023/B:CMAJ.0000024498.19392.a9