A Lyapunov-type stability criterion using norms

Authors:
Meirong Zhang and Weigu Li

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3325-3333

MSC (2000):
Primary 34L15, 34D20, 34C25

DOI:
https://doi.org/10.1090/S0002-9939-02-06462-6

Published electronically:
March 25, 2002

MathSciNet review:
1913012

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a -periodic potential such that . The classical Lyapunov criterion to stability of Hill's equation is , where is the negative part of . In this paper, we will use a relation between the (anti-)periodic and the Dirichlet eigenvalues to establish some lower bounds for the first anti-periodic eigenvalue. As a result, we will find the best Lyapunov-type stability criterion using norms of , . The numerical simulation to Mathieu's equation shows that the new criterion approximates the first stability region very well.

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Additional Information

**Meirong Zhang**

Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China

Email:
mzhang@math.tsinghua.edu.cn

**Weigu Li**

Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Email:
weigu@math.pku.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-02-06462-6

Keywords:
Hill's equation,
Lyapunov stability,
eigenvalue

Received by editor(s):
October 3, 2000

Received by editor(s) in revised form:
June 15, 2001

Published electronically:
March 25, 2002

Additional Notes:
This project was supported by the National Natural Science Foundation of China, The National 973 Project of China, and The Excellent Personnel Supporting Plan of the Ministry of Education of China

Communicated by:
Carmen Chicone

Article copyright:
© Copyright 2002
American Mathematical Society