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

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