Linear transformation and oscillation criteria for Hamiltonian systems. (English) Zbl 1124.34021
The author studies the linear Hamiltonian system
\[
x'=A(t)x+B(t)u, \qquad u'=C(t)x-A^*(t)u,
\]
where \(A(t)\), \(B(t)\), \(C(t)\) are real \(n\times n\) matrix-valued functions, \(B\) and \(C\) are Hermitian, \(B\) is positive definite. Using a transformation which preserves oscillation properties of this system, the author derives new oscillation criteria from known oscillation criteria of other authors.
Reviewer: Robert Mařík (Brno)
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
References:
| [1] | Ahlbrandt, C. D.; Hinton, D. B.; Lewis, R. T., The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory, J. Math. Anal. Appl., 81, 234-277 (1981) · Zbl 0459.34018 |
| [2] | Mingarelli, A. B., Nonlinear functionals in oscillation theory of matrix differential systems, Comm. Pure Appl. Math., 3, 75-84 (2004) · Zbl 1055.34067 |
| [3] | Butler, G. J.; Erbe, L. H.; Mingarelli, A. B., Riccati techniques and variational principles in oscillation theory for linear systems, Tran. Amer. Math. Soc., 303, 263-282 (1987) · Zbl 0648.34031 |
| [4] | Meng, Fanwei; Mingarelli, A. B., Oscillation of linear Hamiltonian systems, Proc. Amer. Math. Soc., 131, 897-904 (2002) · Zbl 1008.37032 |
| [5] | Hinton, D. B.; Lewis, R. T., Oscillation theory of generalized second order differential equations, Rocky Mountain J. Math., 10, 751-761 (1980) · Zbl 0485.34021 |
| [6] | Sowjaya Kumari, I.; Umanaheswaram, S., Oscillation criteria for linear matrix Hamiltonian systems, J. Differential Equations, 165, 174-198 (2000) · Zbl 0970.34025 |
| [7] | Yang, Q.; Mathsen, R.; Zhu, S., Oscillation theorems for self-adjoint matrix Hamiltonian systems, J. Differential Equations, 190, 306-329 (2003) · Zbl 1032.34033 |
| [8] | Chen, Shaozhu; Zheng, Zhaowen, Oscillation criteria of Yan type for linear Hamiltonian systems, Comput. Math. Appl., 46, 855-862 (2003) · Zbl 1049.34038 |
| [9] | Zheng, Zhaowen, Oscillation theorems for linear matrix Hamiltonian systems, Appl. Math. J. Chinese Univ., 17, 285-290 (2002), (in Chinese) · Zbl 1015.34022 |
| [10] | Meng, Fanwei; Sun, Yuangong, Oscillation for linear Hamiltonian systems, Comput. Math. Appl., 44, 1467-1477 (2002) · Zbl 1047.34030 |
| [11] | Byers, R.; Harris, B. J.; Kwong, M. K., Weighted means and oscillation conditions for second order matrix differential systems, J. Differential Equations, 61, 164-177 (1986) · Zbl 0609.34042 |
| [12] | Coles, W. J., Oscillation for self-adjoint second order matrix differential equations, Differential Integral Equations, 4, 195-204 (1991) · Zbl 0722.34026 |
| [13] | 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 |
| [14] | Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math., 7, 115-117 (1949) · Zbl 0032.34801 |
| [15] | Kamenev, I. V., Integral criterion for oscillation of linear differential equations of second order, Mat. Zametki, 23, 249-251 (1978), (in Russian) · Zbl 0386.34032 |
| [16] | Etgen, G. J.; Pawlowski, J. F., A comparison theorem and oscillation criteria for second order differential systems, Pacific J. Math., 72, 59-69 (1977) · Zbl 0373.34016 |
| [17] | 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 |
| [18] | Kwong, M. K.; Zettl, A., Integral inequalities and second order linear oscillation, J. Differential Equations, 46, 16-33 (1982) · Zbl 0498.34022 |
| [19] | Walters, T., A characterization of positive linear functional and oscillation criteria for matrix differential equations, Proc. Amer. Math. Soc., 78, 198-202 (1980) · Zbl 0446.34038 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
