Oscillation of linear Hamiltonian systems

Authors:
Fanwei Meng and Angelo B. Mingarelli

Journal:
Proc. Amer. Math. Soc. **131** (2003), 897-904

MSC (2000):
Primary 34A30, 34C10

Published electronically:
July 25, 2002

MathSciNet review:
1937428

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Abstract | References | Similar Articles | Additional Information

Abstract: We establish new oscillation criteria for linear Hamiltonian systems using monotone functionals on a suitable matrix space. In doing so we develop new criteria for oscillation involving general monotone functionals instead of the usual largest eigenvalue. Our results are new even in the particular case of self-adjoint second order differential systems.

**1.**G. J. Butler and L. H. Erbe,*Oscillation results for second order differential systems*, SIAM J. Math. Anal.**17**(1986), no. 1, 19–29. MR**819208**, 10.1137/0517003**2.**G. J. Butler and L. H. Erbe,*Oscillation results for selfadjoint differential systems*, J. Math. Anal. Appl.**115**(1986), no. 2, 470–481. MR**836240**, 10.1016/0022-247X(86)90009-0**3.**G. J. Butler, L. H. Erbe, and A. B. Mingarelli,*Riccati techniques and variational principles in oscillation theory for linear systems*, Trans. Amer. Math. Soc.**303**(1987), no. 1, 263–282. MR**896022**, 10.1090/S0002-9947-1987-0896022-5**4.**Ralph Byers, B. J. Harris, and Man Kam Kwong,*Weighted means and oscillation conditions for second order matrix differential equations*, J. Differential Equations**61**(1986), no. 2, 164–177. MR**823400**, 10.1016/0022-0396(86)90117-8**5.**W. A. Coppel,*Disconjugacy*, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR**0460785****6.**Lynn H. Erbe, Qingkai Kong, and Shi Gui Ruan,*Kamenev type theorems for second-order matrix differential systems*, Proc. Amer. Math. Soc.**117**(1993), no. 4, 957–962. MR**1154244**, 10.1090/S0002-9939-1993-1154244-0**7.**Philip Hartman,*Self-adjoint, non-oscillatory systems of ordinary, second order, linear differential equations*, Duke Math. J.**24**(1957), 25–35. MR**0082591****8.**Don B. Hinton and Roger T. Lewis,*Oscillation theory for generalized second-order differential equations*, Rocky Mountain J. Math.**10**(1980), no. 4, 751–766. MR**595103**, 10.1216/RMJ-1980-10-4-751**9.**Werner Kratz,*Quadratic functionals in variational analysis and control theory*, Mathematical Topics, vol. 6, Akademie Verlag, Berlin, 1995. MR**1334092****10.**Man Kam Kwong and Hans G. Kaper,*Oscillation of two-dimensional linear second-order differential systems*, J. Differential Equations**56**(1985), no. 2, 195–205. MR**774162**, 10.1016/0022-0396(85)90104-4**11.**Fanwei Meng, Jizhong Wang, and Zhaowen Zheng,*A note on Kamenev type theorems for second order matrix differential systems*, Proc. Amer. Math. Soc.**126**(1998), no. 2, 391–395. MR**1443844**, 10.1090/S0002-9939-98-04248-8**12.**Angelo B. Mingarelli,*On a conjecture for oscillation of second-order ordinary differential systems*, Proc. Amer. Math. Soc.**82**(1981), no. 4, 593–598. MR**614884**, 10.1090/S0002-9939-1981-0614884-3**13.**Allan Peterson and Jerry Ridenhour,*Oscillation of second order linear matrix difference equations*, J. Differential Equations**89**(1991), no. 1, 69–88. MR**1088335**, 10.1016/0022-0396(91)90111-L**14.**William T. Reid,*Sturmian theory for ordinary differential equations*, Applied Mathematical Sciences, vol. 31, Springer-Verlag, New York-Berlin, 1980. With a preface by John Burns. MR**606199**

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Additional Information

**Fanwei Meng**

Affiliation:
Department of Mathematics, Qufu Normal University, Qufu, Shandong, 273165, People’s Republic of China

Email:
fwmeng@qfnu.edu.cn

**Angelo B. Mingarelli**

Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6

DOI:
https://doi.org/10.1090/S0002-9939-02-06614-5

Keywords:
Oscillation,
Hamiltonian systems,
monotone functionals

Received by editor(s):
October 19, 2001

Published electronically:
July 25, 2002

Additional Notes:
This research was supported by the NSF of China (10071043) and Shandong Province (FWM) and NSERC Canada (ABM)

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2002
American Mathematical Society