Some new oscillation results for linear Hamiltonian systems. (English) Zbl 1161.34002

Li, Lianzhong; Meng, Fanwei; Zheng, Zhaowen

Appl. Math. Comput. 208, No. 1, 219-224 (2009).

The authors consider linear Hamiltonian systems of the form \(X^\prime = A(t)X+B(t)Y, Y^\prime=C(t)X-A^\ast (t)Y\), where \(A, B=B^\ast >0, C=C^\ast, X, Y\) are \(n \times n\)-matrices and \(t \in [t_0,\infty)\). It is said that such a system is oscillatory if for any solution \((X,Y)\) such that its Wronskian vanishes: \(X^\ast (t)Y(t) - Y^\ast (t)X(t) =0\), and \(\det X(t_1) \neq 0\) for some \(t_1\in [t_0,\infty)\) the determinant \(X(t)\) has arbitrary large zeros. By using the standard integral average technique they derive some new criteria for such a system to be oscillatory.

Reviewer: Iskander A. Taimanov (Novosibirsk)

MSC:

34A30	Linear ordinary differential equations and systems
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

linear Hamiltonian system; oscillation; integral average

References:

[1]	Bultler, G. J.; Erbe, L. H., Oscillation results for second order differential systems, SIAM J. Math. Anal., 17, 19-29 (1986) · Zbl 0583.34027
[2]	Bultler, G. J.; Erbe, L. H., Oscillation results for self-adjoint differential systems, J. Math. Anal. Appl., 115, 470-481 (1986) · Zbl 0588.34025
[3]	Bultler, G. J.; Erbe, L. H.; Mingarelli, A. B., Riccati techniques and variational principles in oscillation theory for linear systems, Trans. Amer. Math. Soc., 303, 263-282 (1987) · Zbl 0648.34031
[4]	Byers, R.; Harris, B. J.; Kwong, M. K., Weighted means and oscillation conditions for second order matrix differential equations, J. Differential Equations, 61, 164-177 (1986) · Zbl 0609.34042
[5]	Erbe, L. H.; Kong, Q.; Ruan, S., Kamenev type theorems for second order matrix differential systems, Proc. Amer. Math. Soc., 117, 957-962 (1993) · Zbl 0777.34024
[6]	Kwong, M. K.; Kaper, H. G., Oscillation of two-dimensional linear second order differential systems, J. Differential Equations, 56, 195-205 (1985) · Zbl 0571.34024
[7]	Kwong, M. K.; Kaper, H. G.; Akiyama, K., Oscillation of linear second order differential systems, Proc. Amer. Math. Soc., 91, 85-91 (1984) · Zbl 0556.34026
[8]	Meng, F.; Wang, J.; Zheng, Z., A note on Kamenev type theorems for second order matrix differential systems, Proc. Amer. Math. Soc., 126, 391-395 (1998) · Zbl 0891.34037
[9]	Meng, F., Oscillation results for linear Hamiltonian systems, Appl. Math. Comput., 131, 357-372 (2002) · Zbl 1055.34066
[10]	Meng, F.; Sun, Y., Oscillation of linear Hamiltonian systems, Comp. Math. Appl., 44, 1467-1477 (2002) · Zbl 1047.34030
[11]	Sun, Y., New oscillation criteria for liner matrix Hamiltonian systems, J. Math. Anal. Appl., 279, 651-658 (2003) · Zbl 1032.34032
[12]	Sowjanya Kumari, I.; Umamaheswaram, S., Oscillation criteria for liner matrix Hamiltonian systems, J. Differential Equations, 165, 174-189 (2000) · Zbl 0970.34025
[13]	Rickart, C. E., Banach Algebras (1960), Van Nostrand: Van Nostrand New York · Zbl 0051.09106
[14]	Sun, Y.; Meng, F., New oscillation criteria for linear matrix Hamiltonian systems, Appl. Math. Comput., 155, 259-268 (2004) · Zbl 1058.34037
[15]	Chen, S.; Zheng, Z., Oscillation criteria of Yan type for linear Hamiltonian systems, Comput. Math. Appl., 46, 6, 855-862 (2003) · Zbl 1049.34038
[16]	Zheng, Z.; Zhu, S., Oscillatory properties for linear matrix Hamiltonian systems, Dynam. Syst. Appl., 13, 2, 317-326 (2004) · Zbl 1088.34025
[17]	Zheng, Z., Linear transformation and oscillation criteria for Hamiltonian systems, J. Math. Anal. Appl., 332, 236-245 (2007) · Zbl 1124.34021
[18]	Zheng, Z., Interval oscillation criteria for linear Hamiltonian systems, Math. Nachr., 281, 11, 1664-1671 (2008) · Zbl 1161.34003