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Keywords:
delay differential equation; equilibrium; convergence
delay differential equation; equilibrium; convergence
Summary:
Consider the delay differential equation \[ \dot{x}(t)=g(x(t),x(t-r)), \qquad \mathrm{(1)}\] where $r>0$ is a constant and $g\:\mathbb{R}^2\rightarrow \mathbb{R}$ is Lipschitzian. It is shown that if $r$ is small, then the solutions of (1) have the same convergence properties as the solutions of the ordinary differential equation obtained from (1) by ignoring the delay.
References:
[1] O. Arino, M. Pituk: More on linear differential systems with small delays. J. Differ. Equations 170 (2001), 381–407. DOI 10.1006/jdeq.2000.3824 | MR 1815189
[2] R. D. Driver: Linear differential systems with small delays. J. Differ. Equations 21 (1976), 149–167. MR 0404803 | Zbl 0319.34067
[3] I. Györi: Interaction between oscillations and global asymptotic stability in delay differential equations. Differ. Integral Equ. 3 (1990), 181–200. MR 1014735
[4] I. Györi, M. Pituk: Stability criteria for linear delay differential equations. Differ. Integral Equ. 10 (1997), 841–852. MR 1741755
[5] I. Györi, M. Pituk: Special solutions of neutral functional differential equations. J. Inequal. Appl. 6 (2001), 99–117. MR 1887327
[6] J. Hale: Theory of Functional Differential Equations. Springer, New York, 1977. MR 0508721 | Zbl 0352.34001
[7] J. Jarník, J. Kurzweil: Ryabov’s special solutions of functional differential equations. Boll. Un. Mat. Ital. 11 (1975), 198–218. MR 0454264
[8] T. Krisztin, H.-O. Walther, J. Wu: Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Positive Feedback. Fields Institute Monograph Series, Vol. 11, Amer. Math. Soc., Providence, RI, 1999. MR 1719128
[9] M. Pituk: Convergence to equilibria in scalar non-quasi-monotone functional differential equations. In preparation.
[10] Yu. A. Ryabov: Certain asymptotic properties of linear systems with small time lag. Trudy Sem. Teor. Differencial. Uravnenii s Otklon. Argumentom Univ. Druzby Narodov Patrica Lumumby 3 (1965), 153–164. (Russian) MR 0211010
[11] H. L. Smith: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Amer. Math. Soc., Providence, RI, 1995. MR 1319817 | Zbl 0821.34003
[12] H. L. Smith, H. Thieme: Monotone semiflows in scalar non-quasi-monotone functional differential equations. J. Math. Anal. Appl. 150 (1990), 289–306. DOI 10.1016/0022-247X(90)90105-O | MR 1067429
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