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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Gauge functions and limit sets for nonautonomous ordinary differential equations


Author: H. K. Wilson
Journal: Proc. Amer. Math. Soc. 35 (1972), 487-490
MSC: Primary 34C05
DOI: https://doi.org/10.1090/S0002-9939-1972-0303004-5
MathSciNet review: 0303004
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Abstract: A gauge function V for the differential equation $ ({\text{S}})x' = f(x,t)$ is a scalar-valued function sufficiently smooth for $ dV(\phi (t),t)/dt$ to exist almost everywhere for solutions $ x = \phi (t)$, $ {t_0} \leqq t < \tau _\phi ^ + $, of (S). Let (S) have gauge function V that satisfies the following conditions: (1) $ {\lim _{t \to + \infty }}V(x,t) \equiv \lambda (x)$ exists; (2) V is continuous in x uniformly with respect to t; (3) the upper, right derivate of V with respect to (S) is nonpositive. Then, if a solution $ x = \phi (t)$ of (S) has an $ \omega $-limit point P, there is a unique constant $ c(\phi )$ such that $ \lambda (p) = c(\phi )$. An application to second order, linear equations is given.


References [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] H. K. Wilson, Ordinary differential equations. Introductory and intermediate courses using matrix methods, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1971. MR 0280764

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0303004-5
Keywords: Gauge function, $ \omega $-limit set, positive limit set, asymptotic behavior of solutions, linear second order equation
Article copyright: © Copyright 1972 American Mathematical Society

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