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Lyapunov inequalities and stability for linear Hamiltonian systems. (English) Zbl 1242.37039

The paper considers qualitative properties of the Hamiltonian system

Ju ' =H(t)u,u= col (x T y T ) n

expressed via various Lyapunov-like inequalities. Discussed are the standard boundary value problem


and its applications to quantum mechanics. The paper ends with some stability criteria for planar Hamiltonian systems. All results can be seen as either improvements or generalizations of Lyapunov inequalities.

37J25Stability problems (finite-dimensional Hamiltonian etc. systems)
34B05Linear boundary value problems for ODE
34D20Stability of ODE
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
[1]Cabada, A.; Cid, J. A.; Tvrdy, M.: A generalized anti-maximum principle for the periodic one dimensional p-Laplacian with sign changing potential, Nonlinear anal. 72, 3436-3446 (2010) · Zbl 1192.34025 · doi:10.1016/j.na.2009.12.028
[2]Cañada, A.; Villegas, S.: Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using lp norms, Discrete contin. Dyn. syst. 20, 877-888 (2008) · Zbl 1162.34011 · doi:10.3934/dcds.2008.20.877
[3]Cañada, A.; Villegas, S.: Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues, J. eur. Math. soc. 12, 163-178 (2010) · Zbl 1201.34024 · doi:10.4171/JEMS/193
[4]Cañada, A.; Villegas, S.: Matrix Lyapunov inequalities for ordinary and elliptic partial differential equations
[5]Cheng, S. S.: Lyapunov inequalities for differential and difference equations, Fasc. math. 23, 25-41 (1991) · Zbl 0753.34017
[6]Chu, J.; Zhang, M.: Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete contin. Dyn. syst. 21, 1071-1094 (2008) · Zbl 1161.37041 · doi:10.3934/dcds.2008.21.1071 · doi:http://aimsciences.org/journals/pdfs.jsp?paperID=3356{&}mode=abstract
[7]Clark, S.; Hinton, D.: Some disconjugacy criteria for differential equations with oscillatory coefficients, Math. nachr. 278, 1476-1489 (2005) · Zbl 1101.34018 · doi:10.1002/mana.200410316
[8]Coddington, E. A.; Levinson, N.: Stability and asymptotic behavior of differential equations, (1965)
[9]Coppel, W. A.; Heath, D. C.: Theory of ordinary differential equations, (1955)
[10]Das, A. M.; Vatsala, A. S.: Green function for n-n boundary value problem and an analogue of hartman’s result, J. math. Anal. appl. 51, 670-677 (1975) · Zbl 0312.34011 · doi:10.1016/0022-247X(75)90117-1
[11]Guseinov, G. Sh.; Kaymakcalan, B.: Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. math. Appl. 45, 1399-1416 (2003) · Zbl 1055.39029 · doi:10.1016/S0898-1221(03)00095-6
[12]Guseinov, G. Sh.; Zafer, A.: Stability criteria for linear periodic impulsive Hamiltonian systems, J. math. Anal. appl. 335, 1195-1206 (2007) · Zbl 1128.34005 · doi:10.1016/j.jmaa.2007.01.095
[13]Hartman, P.: Ordinary differential equations, (1982) · Zbl 0476.34002
[14]Kong, Q.; Zettl, A.: Eigenvalues of regular Sturm-Liouville problems, J. differential equations 131, 1-19 (1996) · Zbl 0862.34020 · doi:10.1006/jdeq.1996.0154
[15]Krein, M. G.: Foundations of the theory of λ-zones of stability of canonical system of linear differential equations with periodic coefficients, Amer. math. Soc. transl. Ser. 2 120, 413-498 (1955) · Zbl 0516.34049
[16]Kunze, M.; Ortega, R.: On the number of solutions to semilinear boundary value problems, Adv. nonlinear stud. 4, 237-249 (2004) · Zbl 1142.35425
[17]Lerin, A. J.: Some problems bearing on the oscillation solutions of linear differential equations, Soviet math. Dokl. 5, 818-821 (1964)
[18]Levitan, B. M.; Sargsjan, L. S.: Sturm-Liouville and Dirac operators, Math. appl. (Soviet ser.) 59 (1991)
[19]Lyapunov, A. M.: Problème général de la stabilité du mouvement, Ann. fac. Sci. univ. Toulouse 2 9, 203-474 (1907)
[20]Meng, G.; Yan, P.; Lin, X.; Zhang, M.: Non-degeneracy and periodic solutions of semilinear differential equations with deviation, Adv. nonlinear stud. 6, 563-590 (2006) · Zbl 1120.34051
[21]Meng, G.; Zhang, M.: Continuity in weak topology: first order linear systems of ODE, Acta math. Sin. (Engl. Ser.) 26, 1287-1298 (2010) · Zbl 1213.34015 · doi:10.1007/s10114-010-8103-x
[22]Möller, M.; Zettl, A.: Differentiable dependence of eigenvalues of operators in Banach spaces, J. operator theory 36, 335-355 (1996) · Zbl 0869.47004
[23]Reid, W. T.: A matrix Liapunov inequality, J. math. Anal. appl. 32, 424-434 (1970) · Zbl 0208.11303 · doi:10.1016/0022-247X(70)90308-2
[24]Reid, W. T.: A generalized Liapunov inequality, J. differential equations 13, 182-196 (1973) · Zbl 0248.34043 · doi:10.1016/0022-0396(73)90040-5
[25]Reid, W. T.: Interrelations between a trace formula and Liapunov inequalities, J. differential equations 23, 448-458 (1977) · Zbl 0345.34027 · doi:10.1016/0022-0396(77)90122-X
[26]Siegel, C.; Moser, J.: Lectures on celestial mechanics, (1971)
[27]Tang, X. H.; Zhou, Y. G.: Periodic solutions of a class of nonlinear functional differential equations and global attractivity, Acta math. Sinica 49, 899-908 (2006) · Zbl 1124.34350
[28]Vidyasagar, M.: Nonlinear system analysis, (1978)
[29]Wang, X.: Stability criteria for linear periodic Hamiltonian systems, J. math. Anal. appl. 367, 329-336 (2010) · Zbl 1195.34079 · doi:10.1016/j.jmaa.2010.01.027
[30]Yan, P.; Zhang, M.: Continuity in weak topology and extremal problems of eigenvalues of the p-Laplacian, Trans. amer. Math. soc. 363, 2003-2028 (2011) · Zbl 1221.47107 · doi:10.1090/S0002-9947-2010-05051-2
[31]Zettl, A.: Sturm-Liouville theory, Math. surveys monogr. 121 (2005) · Zbl 1103.34001
[32]Zhang, M.: Certain classes of potentials for p-Laplacian to be non-degenerate, Math. nachr. 278, 1823-1836 (2005) · Zbl 1092.34044 · doi:10.1002/mana.200410342
[33]Zhang, M.: Sobolev inequalities and ellipticity of planar linear Hamiltonian systems, Adv. nonlinear stud. 8, 633-654 (2008) · Zbl 1165.34053
[34]Zhang, M.: From ODE to DDE, Front. math. China 4, 585-598 (2009) · Zbl 1188.34093 · doi:10.1007/s11464-009-0034-4
[35]Zhang, M.: Optimal criteria for maximum and antimaximum principles of the periodic solution problem, Bound. value probl. 2010 (2010) · Zbl 1200.34001 · doi:10.1155/2010/410986
[36]Zhang, M.; Li, W.: A Lyapunov-type stability criterion using Lα norms, Proc. amer. Math. soc. 130, 3325-3333 (2002) · Zbl 1007.34053 · doi:10.1090/S0002-9939-02-06462-6