New Kamenev type oscillation criteria for linear matrix Hamiltonian systems. (English) Zbl 1060.34018

Sun, Yuan Gong

Appl. Math. Comput. 158, No. 1, 69-78 (2004).

Summary: Some new Kamenev-type criteria are obtained for the oscillation of the linear matrix Hamiltonian system \[ X'=A(t)X+ B(t)Y,\quad Y'= C(t)X- A^*(t)Y, \] under the hypothesis: \(A(t)\), \(B(t)= B^*(t)> 0\) and \(C(t)= C^*(t)\) are real continuous \(n\times n\)-matrix functions on the interval \([t_0,\infty)\), \(t_0>-\infty\). Our results are different from most known ones in the sense that they are given in the form of \(\lim_{t\to\infty}\sup g[\cdot]>\text{const.}\), rather than in the form of \(\lim_{t\to\infty}\sup \lambda_1[\cdot]= \infty\), where \(g\) is a positive linear functional on the linear space of \(n\times n\)-matrices with real entries. Consequently, our results improve some previous results to some extent, which can be seen by the examples given at the end of this paper.

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

Oscillation; Linear Hamiltonian systems; Positive linear functional

References:

[1]	Butler, G. J.; Erbe, L. H., Oscillation results for second order differential systems, SIAM J. Math. Anal, 17, 19-29 (1986) · Zbl 0583.34027
[2]	Butler, G. J.; Erbe, L. H., Oscillation results for self-adjoint differential systems, J. Math. Anal. Appl, 115, 470-481 (1986) · Zbl 0588.34025
[3]	Butler, 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. Different. Eqs, 61, 164-177 (1986) · Zbl 0609.34042
[5]	Coppel, W. A., Disconjugacy, Lecture notes in Mathematics, vol. 220 (1971), Springer: Springer Berlin · Zbl 0224.34003
[6]	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
[7]	Hartman, P., Self-adjoint, non-oscillatory systems of ordinary second order, linear differential equations, Duke Math. J, 24, 25-36 (1957) · Zbl 0077.08701
[8]	Hinton, D. B.; Lewis, R. T., Oscillation theory for generalized second order differential equations, Rocky Mountain J. Math, 10, 751-766 (1980) · Zbl 0485.34021
[9]	Sowjaya Kumari, I.; Umanaheswaram, S., Oscillation criteria for linear matrix Hamiltonian systems, J. Different. Eqs, 165, 174-198 (2000) · Zbl 0970.34025
[10]	Kwong, M. K.; Kaper, H. G., Oscillation of two-dimensional linear second order differential systems, J. Different. Eqs, 56, 195-205 (1985) · Zbl 0571.34024
[11]	Kwong, M. K.; Kaper, H. G.; Akiyama, K.; Mingarelli, A. B., Oscillation of linear second-order differential systems, Proc. Amer. Math. Soc, 91, 85-91 (1984) · Zbl 0556.34026
[12]	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
[13]	Meng, F. W.; Sun, Y. G., Oscillation of linear Hamiltonian systems, Comp. Math. Appl, 44, 1467-1477 (2002) · Zbl 1047.34030
[14]	Sun, Y. G., New oscillation criteria for linear matrix Hamiltonian systems, J. Math. Anal. Appl, 279, 651-658 (2003) · Zbl 1032.34032
[15]	Mingarelli, A. B., On a conjecture for oscillation of second order ordinary differential systems, Proc. Amer. Math. Soc, 82, 593-598 (1981) · Zbl 0487.34030
[16]	Rickart, C. E., Banach Algebras (1960), Van Nostrand: Van Nostrand New York · Zbl 0051.09106