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Oscillation theorems for self-adjoint matrix Hamiltonian systems involving general means. (English) Zbl 1058.34039
By employing monotone functionals, a generalized Riccati transformation and the general means technique, new oscillation criteria are obtained for the selfadjoint matrix Hamiltonian system $X'=A(t)X+B(t)Y,\quad Y'=C(t)X-A^*(t)Y$. These criteria improve and complement a number of existing results.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A30 Linear ODE and systems, general 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
##### Keywords:
matrix Hamiltonian system; oscillation
Full Text:
##### References:
 [1] Butler, G. J.; Erbe, L. H.; Mingarelli, A. B.: Riccati techniques and variational principles in oscillatory theory for linear system. Trans. amer. Math. soc. 303, 263-282 (1987) · Zbl 0648.34031 [2] Byers, R.; Harris, B. J.; Kwong, M. K.: Weighted means and oscillation of second order matrix differential system. J. differential equations 61, 164-177 (1986) · Zbl 0609.34042 [3] Coles, W. J.; Kinyon, M. K.: Summability methods for oscillation of linear second order matrix differential equation. Rocky mountain J. Math. 24, 19-36 (1994) · Zbl 0807.34042 [4] 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 [5] Hartman, P.: On nonoscillatory linear differential equation of second order. Amer. J. Math. 74, 389-400 (1952) · Zbl 0048.06602 [6] Hartman, P.: Oscillation criteria for self-adjoint second-order differential systems and ”principal sectional curvatures”. J. differential equations 34, 326-338 (1979) · Zbl 0443.34029 [7] Li, H. J.: Oscillation criteria for second order linear differential equations. J. math. Anal. appl. 194, 217-234 (1995) · Zbl 0836.34033 [8] Kamenev, I. V.: An integral criteria for oscillation of linear differential equations. Mat. zametki 23, 249-251 (1978) · Zbl 0386.34032 [9] Kratz, W.: Oscillation criteria theorem for self-adjoint differential systems and an index results for corresponding matrix differential equations. Math. proc. Philos. soc. 118, 351-361 (1995) · Zbl 0847.34036 [10] Kumari, I. S.; Umamaheswaram, S.: Oscillation criteria for linear matrix Hamiltonian systems. J. differential equations 165, 165-174 (2000) · Zbl 0970.34025 [11] Meng, F.: Oscillation results for linear Hamiltonian systems. Appl. math. Comput. 131, 357-372 (2002) · Zbl 1055.34066 [12] Meng, F.; Wang, J.; Zheng, Z.: A note on kamenev type theorem for second order matrix differential systems. Proc. amer. Math. soc. 126, 391-395 (1998) · Zbl 0891.34037 [13] Philos, Ch.G: On a kamenev’s integral criteria for oscillation of linear differential equations of second order. Util. math. 24, 277-289 (1983) · Zbl 0528.34035 [14] Philos, Ch.G: Oscillation theorems for linear differential equations of second order. Arch. math. 53, 483-492 (1989) · Zbl 0661.34030 [15] Reid, W. T.: Sturmian theory for ordinary differential equations. Appl. math. Sci. 31 (1980) · Zbl 0459.34001 [16] Rickart, C. E.: Banach algebras. (1960) · Zbl 0095.09702 [17] Wang, Q.: Oscillation criteria for second order matrix differential systems. Arch. math. 76, 385-390 (2001) · Zbl 0989.34024 [18] Wintner, A.: A criteria of oscillatory stability. Quart. appl. Math. 7, 115-117 (1949) · Zbl 0032.34801 [19] Wong, James S. W.: On kamenev-type oscillation criteria for second order differential equations with damping. J. math. Anal. appl. 258, 244-257 (2001) · Zbl 0987.34024 [20] Yang, Q. G.; Mathsen, R.; Zhu, S. M.: Oscillation theorems for self-adjoint matrix Hamiltonian systems. J. differential equations 190, 306-329 (2003) · Zbl 1032.34033 [21] Yang, Q. G.: Oscillation theorems for second order linear self-adjoint matrix differential systems with damping. Acta math. Sinica 20 (2004)