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March  2012, 11(2): 465-473. doi: 10.3934/cpaa.2012.11.465

Lyapunov-type inequalities for even order differential equations

1. 

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received  January 2011 Revised  May 2011 Published  October 2011

In this paper, we establish several new Lyapunov-type inequalities for the $2n-$order differential equation

$x^{(2n)}(t)+(-1)^{n-1}q(t)x(t)=0, $

which are sharper than all related existing ones.

Citation: Xiaofei He, X. H. Tang. Lyapunov-type inequalities for even order differential equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 465-473. doi: 10.3934/cpaa.2012.11.465
References:
[1]

S. S. Cheng, A discrete analogue of the inequality of Lyapunov,, Hokkaido Math. J., 12 (1983), 105.  doi: /327/1/HMJ12-105.  Google Scholar

[2]

S. S. Cheng, Lyapunov inequalities for differential and difference equations,, Hokkaido Math. Fasc. Math., 23 (1991), 25.   Google Scholar

[3]

D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations,, Appl. Math. Comput., 216 (2010), 368.  doi: 10.1016/j.amc.2010.01.010.  Google Scholar

[4]

K. M. Das and A. S. Vatsala, Green's function for n-n boundary value problemand an analogue of Hartman's result,, J. Math. Anal. Appl., 51 (1975), 670.  doi: 10.1016/0022-247X(75)90117-1.  Google Scholar

[5]

S. B. Eliason, A Lyapunov inequality for a certain second order nonlinear differential equation,, J. London Math. Soc., 2 (1970), 461.   Google Scholar

[6]

S. B. Eliason, Lyapunov type inequalities for certain second order functional differential equations,, SIAM J. Appl. Math., 27 (1974), 180.  doi: 10.1137/0127015.  Google Scholar

[7]

G. Sh. Guseinov and B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems,, Comput. Math. Appl., 45 (2003), 1399.  doi: 10.1016/S0898-1221(03)00095-6.  Google Scholar

[8]

G. Sh. Guseinov and A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems,, J. Math. Anal. Appl., 335 (2007), 1195.  doi: 10.1016/j.jmaa.2007.01.095.  Google Scholar

[9]

P. Hartman and A. Wintner, On an oscillation criterion of Lyapunov,, Amer. J. Math., 73 (1951), 885.  doi: jstor.org/stable/2372122.  Google Scholar

[10]

H. Hochstadt, A new proof of a stability estimate of Lyapunov,, Proc. Amer. Math. Soc., 14 (1963), 525.  doi: 10.1090/S0002-9939-1963-0149019-6.  Google Scholar

[11]

L. Q. Jiang and Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales,, J. Math. Anal. Appl., 310 (2005), 579.  doi: 10.1016/j.jmaa.2005.02.026.  Google Scholar

[12]

M. K. Kwong, On Lyapunov's inequality for disfocality,, J. Math. Anal. Appl., 83 (1981), 486.  doi: 10.1016/0022-247X(81)90137-2.  Google Scholar

[13]

C. Lee, C. Yeh, C. Hong and R. P. Agarwal, Lyapunov and Wirtinger inequalities,, Appl. Math. Lett., 17 (2004), 847.  doi: 10.1016/j.aml.2004.06.016.  Google Scholar

[14]

A. M. Liapunov, Problème général de la stabilité du mouvement,, Fac. Sci. Univ. Toulouse., 2 (1907), 203.   Google Scholar

[15]

Z. Nehari, Some eigenvalue estimates,, J. D'analyse Math., 7 (1959), 79.  doi: 10.1007/BF02787681.  Google Scholar

[16]

Z. Nehari, "On an inequality of Lyapunov, Studies in Mathematical Analysis and Related Topics,", Stanford University Press, (1962).   Google Scholar

[17]

B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations,, J. Math. Anal. Appl., 195 (1995), 527.  doi: 10.1006/jmaa.1995.1372.  Google Scholar

[18]

J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-Laplacian,, Abstr. Appl. Anal., 2004 (2004), 147.  doi: 10.1155/S108533750431002X.  Google Scholar

[19]

T. W. Reid, A matrix equation related to a non-oscillation criterion and Lyapunov stability,, Quart. Appl. Math. Soc., 23 (1965), 83.   Google Scholar

[20]

T. W. Reid, A matrix Lyapunov inequality,, J. Math. Anal. Appl., 32 (1970), 424.  doi: 10.1016/0022-247X(70)90308-2.  Google Scholar

[21]

B. Singh, Forced oscillations in general ordinary differential equations,, Tamkang Math. J., 6 (1976), 7.  doi: euclid.hmj/1206135207.  Google Scholar

[22]

X. H. Tang and M. Zhang, Lyapunov inequalities and stability for linear Hamiltonian systems,, J. Differential Equations, In press ().  doi: 10.1016/j.jde.2011.08.002.  Google Scholar

[23]

A. Tiryaki, M. Ünal and D. Cakmak, Lyapunov-type inequalities for nonlinear systems,, J. Math. Anal. Appl., 332 (2007), 497.  doi: 10.1016/j.jmaa.2006.10.010.  Google Scholar

[24]

X. Wang, Stability criteria for linear periodic Hamiltonian systems,, J. Math. Anal. Appl., 367 (2010), 329.  doi: 10.1016/j.jmaa.2010.01.027.  Google Scholar

[25]

X. Yang, On inequalities of Lyapunov type,, Appl. Math. Comput., 134 (2003), 293.  doi: 10.1016/S0096-3003(01)00283-1.  Google Scholar

[26]

X. Yang, On Liapunov-type inequality for certain higher-order differential equations,, Appl. Math. Comput., 134 (2003), 307.  doi: 10.1016/S0096-3003(01)00285-5.  Google Scholar

show all references

References:
[1]

S. S. Cheng, A discrete analogue of the inequality of Lyapunov,, Hokkaido Math. J., 12 (1983), 105.  doi: /327/1/HMJ12-105.  Google Scholar

[2]

S. S. Cheng, Lyapunov inequalities for differential and difference equations,, Hokkaido Math. Fasc. Math., 23 (1991), 25.   Google Scholar

[3]

D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations,, Appl. Math. Comput., 216 (2010), 368.  doi: 10.1016/j.amc.2010.01.010.  Google Scholar

[4]

K. M. Das and A. S. Vatsala, Green's function for n-n boundary value problemand an analogue of Hartman's result,, J. Math. Anal. Appl., 51 (1975), 670.  doi: 10.1016/0022-247X(75)90117-1.  Google Scholar

[5]

S. B. Eliason, A Lyapunov inequality for a certain second order nonlinear differential equation,, J. London Math. Soc., 2 (1970), 461.   Google Scholar

[6]

S. B. Eliason, Lyapunov type inequalities for certain second order functional differential equations,, SIAM J. Appl. Math., 27 (1974), 180.  doi: 10.1137/0127015.  Google Scholar

[7]

G. Sh. Guseinov and B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems,, Comput. Math. Appl., 45 (2003), 1399.  doi: 10.1016/S0898-1221(03)00095-6.  Google Scholar

[8]

G. Sh. Guseinov and A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems,, J. Math. Anal. Appl., 335 (2007), 1195.  doi: 10.1016/j.jmaa.2007.01.095.  Google Scholar

[9]

P. Hartman and A. Wintner, On an oscillation criterion of Lyapunov,, Amer. J. Math., 73 (1951), 885.  doi: jstor.org/stable/2372122.  Google Scholar

[10]

H. Hochstadt, A new proof of a stability estimate of Lyapunov,, Proc. Amer. Math. Soc., 14 (1963), 525.  doi: 10.1090/S0002-9939-1963-0149019-6.  Google Scholar

[11]

L. Q. Jiang and Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales,, J. Math. Anal. Appl., 310 (2005), 579.  doi: 10.1016/j.jmaa.2005.02.026.  Google Scholar

[12]

M. K. Kwong, On Lyapunov's inequality for disfocality,, J. Math. Anal. Appl., 83 (1981), 486.  doi: 10.1016/0022-247X(81)90137-2.  Google Scholar

[13]

C. Lee, C. Yeh, C. Hong and R. P. Agarwal, Lyapunov and Wirtinger inequalities,, Appl. Math. Lett., 17 (2004), 847.  doi: 10.1016/j.aml.2004.06.016.  Google Scholar

[14]

A. M. Liapunov, Problème général de la stabilité du mouvement,, Fac. Sci. Univ. Toulouse., 2 (1907), 203.   Google Scholar

[15]

Z. Nehari, Some eigenvalue estimates,, J. D'analyse Math., 7 (1959), 79.  doi: 10.1007/BF02787681.  Google Scholar

[16]

Z. Nehari, "On an inequality of Lyapunov, Studies in Mathematical Analysis and Related Topics,", Stanford University Press, (1962).   Google Scholar

[17]

B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations,, J. Math. Anal. Appl., 195 (1995), 527.  doi: 10.1006/jmaa.1995.1372.  Google Scholar

[18]

J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-Laplacian,, Abstr. Appl. Anal., 2004 (2004), 147.  doi: 10.1155/S108533750431002X.  Google Scholar

[19]

T. W. Reid, A matrix equation related to a non-oscillation criterion and Lyapunov stability,, Quart. Appl. Math. Soc., 23 (1965), 83.   Google Scholar

[20]

T. W. Reid, A matrix Lyapunov inequality,, J. Math. Anal. Appl., 32 (1970), 424.  doi: 10.1016/0022-247X(70)90308-2.  Google Scholar

[21]

B. Singh, Forced oscillations in general ordinary differential equations,, Tamkang Math. J., 6 (1976), 7.  doi: euclid.hmj/1206135207.  Google Scholar

[22]

X. H. Tang and M. Zhang, Lyapunov inequalities and stability for linear Hamiltonian systems,, J. Differential Equations, In press ().  doi: 10.1016/j.jde.2011.08.002.  Google Scholar

[23]

A. Tiryaki, M. Ünal and D. Cakmak, Lyapunov-type inequalities for nonlinear systems,, J. Math. Anal. Appl., 332 (2007), 497.  doi: 10.1016/j.jmaa.2006.10.010.  Google Scholar

[24]

X. Wang, Stability criteria for linear periodic Hamiltonian systems,, J. Math. Anal. Appl., 367 (2010), 329.  doi: 10.1016/j.jmaa.2010.01.027.  Google Scholar

[25]

X. Yang, On inequalities of Lyapunov type,, Appl. Math. Comput., 134 (2003), 293.  doi: 10.1016/S0096-3003(01)00283-1.  Google Scholar

[26]

X. Yang, On Liapunov-type inequality for certain higher-order differential equations,, Appl. Math. Comput., 134 (2003), 307.  doi: 10.1016/S0096-3003(01)00285-5.  Google Scholar

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