# zbMATH — the first resource for mathematics

New Kamenev type oscillation criteria for linear matrix Hamiltonian systems. (English) Zbl 1060.34018
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
Full Text:
##### 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 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 New York · Zbl 0051.09106
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.