Oscillation of linear Hamiltonian systems

Meng, Fanwei; Mingarelli, Angelo

doi:10.1090/S0002-9939-02-06614-5

Oscillation of linear Hamiltonian systems
HTML articles powered by AMS MathViewer

by Fanwei Meng and Angelo B. Mingarelli PDF

Proc. Amer. Math. Soc. 131 (2003), 897-904 Request permission

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.

References

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, DOI 10.1137/0517003
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, DOI 10.1016/0022-247X(86)90009-0
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, DOI 10.1090/S0002-9947-1987-0896022-5
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, DOI 10.1016/0022-0396(86)90117-8
W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR 0460785
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, DOI 10.1090/S0002-9939-1993-1154244-0
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
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, DOI 10.1216/RMJ-1980-10-4-751
Werner Kratz, Quadratic functionals in variational analysis and control theory, Mathematical Topics, vol. 6, Akademie Verlag, Berlin, 1995. MR 1334092
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, DOI 10.1016/0022-0396(85)90104-4
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, DOI 10.1090/S0002-9939-98-04248-8
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, DOI 10.1090/S0002-9939-1981-0614884-3
Allan Peterson and Jerry Ridenhour, Oscillation of second order linear matrix difference equations, J. Differential Equations 89 (1991), no. 1, 69–88. MR 1088335, DOI 10.1016/0022-0396(91)90111-L
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