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 | References | Similar Articles | Additional Information

Abstract: A gauge function *V* for the differential equation is a scalar-valued function sufficiently smooth for to exist almost everywhere for solutions , , of (S). Let (S) have gauge function *V* that satisfies the following conditions: (1) 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 of (S) has an -limit point *P*, there is a unique constant such that . An application to second order, linear equations is given.

**[1]**T. Yoshizawa,*Stability theory by Liapunov's second method*, Publ. Math. Soc. Japan, no. 9, Math. Soc. Japan, Tokyo, 1966, p. 52. MR**34**#7896. MR**0208086 (34:7896)****[2]**H. K. Wilson,*Ordinary differential equations*, Addison-Wesley, Reading, Mass., 1971, pp. 280-282. MR**43**#6483. MR**0280764 (43:6483)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0303004-5

Keywords:
Gauge function,
-limit set,
positive limit set,
asymptotic behavior of solutions,
linear second order equation

Article copyright:
© Copyright 1972
American Mathematical Society