Oscillation criteria related to integral averaging technique for linear matrix Hamiltonian systems. (English) Zbl 1058.34038

Wang, Qi-Ru

J. Math. Anal. Appl. 295, No. 1, 40-54 (2004).

New oscillation criteria are established for the linear matrix Hamiltonian system \[ U'=A(x)U+B(x)V,\quad V'=C(x)U-A^*(x)V, \tag{*} \] where \(A(x)\), \(B(x)=B^*(x)>0\) and \(C(x)=C^*(x)\) are \(n\times n\)-matrices of real-valued continuous functions on the interval \(I=[x_0,\infty)\). These results are obtained by using the generalized Riccati technique and the integral averaging technique introduced by Ch. G. Philos (1989), and they generalize a number of existing results, e.g., results of L. H. Erbe, Q. Kong and S. Ruan [Proc. Am. Math. Soc. 117, No. 4, 957–962 (1993; Zbl 0777.34024)], I. Sowjanya Kumari and S. Umamaheswaram [J. Differ. Equations 165, No. 1, 174–198 (2000; Zbl 0970.34025)], H. J. Li [J. Math. Anal. Appl. 194, 217–234 (1995; Zbl 0836.34033)] and Z. Zheng, F. Meng and Y. Yu [Acta Math. Sin. 41, No. 6, 1231–1238 (1998; Zbl 1018.34033)]. Three examples illustrate various oscillation criteria.

Reviewer: Zuzana Došlá (Brno)

MSC:

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

Keywords:

matrix Hamiltonian system; oscillation; generalized Riccati technique; integral averaging technique

Citations:

Zbl 0777.34024; Zbl 0970.34025; Zbl 1018.34033; Zbl 0836.34033

References:

[1]	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
[2]	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
[3]	Coles, W. J.; Kinyon, M. K., Summability methods for oscillation of linear second-order matrix differential equations, 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]	Etgen, G. J.; Pawlowski, J. F., Oscillation criteria for second order self-adjoint differential systems, Pacific J. Math., 66, 99-110 (1976) · Zbl 0355.34017
[6]	Hartman, P., On nonscillatory linear differential equations of second order, Amer. J. Math., 74, 389-400 (1952) · Zbl 0048.06602
[7]	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
[8]	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
[9]	Kamenev, I. V., An integral criterion for oscillation of linear differential equations of second order, Mat. Zametki, 23, 249-251 (1978) · Zbl 0386.34032
[10]	Sowjanya Kumari, I.; Umamaheswaram, S., Oscillation criteria for linear matrix Hamiltonian system, J. Differential Equations, 165, 174-198 (2000) · Zbl 0970.34025
[11]	Li, H. J., Oscillation criteria for second order linear differential equations, J. Math. Anal. Appl., 194, 217-234 (1995) · Zbl 0836.34033
[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]	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
[14]	Parhi, N.; Praharaj, P., Oscillation of linear second order matrix differential equations, J. Math. Anal. Appl., 221, 287-305 (1998) · Zbl 0910.34042
[15]	Philos, Ch. G., Oscillation theorems for linear differential equations of second order, Arch. Math. (Basel), 53, 482-492 (1989) · Zbl 0661.34030
[16]	Reid, W. T., Applied Mathematical Sciences (1980), Springer-Verlag: Springer-Verlag New York, vol. 31
[17]	Rogovchenko, Yu. V., Note on ‘Oscillation criteria for second order linear differential equations’, J. Math. Anal. Appl., 203, 560-563 (1996) · Zbl 0862.34024
[18]	Wang, Q. R., Oscillation criteria for second order matrix differential systems, Arch. Math. (Basel), 76, 385-390 (2001) · Zbl 0989.34024
[19]	Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math., 7, 115-117 (1949) · Zbl 0032.34801
[20]	Zheng, Z.; Meng, F.; Yu, Y., On the oscillation of second order linear matrix differential systems, Acta Math. Sinica, 41, 1231-1238 (1998) · Zbl 1018.34033