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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

Xiaofei He 1, and  X. H. Tang 1,

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.

Keywords: Even order, differential equation, Lyapunov-type inequality..
Mathematics Subject Classification: 34B1.
Citation: Xiaofei He, X. H. Tang. Lyapunov-type inequalities for even order differential equations. Communications on Pure and 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), 105-112. doi: /327/1/HMJ12-105.

[2]

S. S. Cheng, Lyapunov inequalities for differential and difference equations, Hokkaido Math. Fasc. Math., 23 (1991), 25-41.

[3]

D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368-373. doi: 10.1016/j.amc.2010.01.010.

[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), 670-677. doi: 10.1016/0022-247X（75）90117-1.

[5]

S. B. Eliason, A Lyapunov inequality for a certain second order nonlinear differential equation, J. London Math. Soc., 2 (1970), 461-466.

[6]

S. B. Eliason, Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math., 27 (1974), 180-199. doi: 10.1137/0127015.

[7]

G. Sh. Guseinov and B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl., 45 (2003), 1399-1416. doi: 10.1016/S0898-1221(03)00095-6.

[8]

G. Sh. Guseinov and A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems, J. Math. Anal. Appl., 335 (2007), 1195-1206. doi: 10.1016/j.jmaa.2007.01.095.

[9]

P. Hartman and A. Wintner, On an oscillation criterion of Lyapunov, Amer. J. Math., 73 (1951), 885-890. doi: jstor.org/stable/2372122.

[10]

H. Hochstadt, A new proof of a stability estimate of Lyapunov, Proc. Amer. Math. Soc., 14 (1963), 525-526. doi: 10.1090/S0002-9939-1963-0149019-6.

[11]

L. Q. Jiang and Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl., 310 (2005), 579-593. doi: 10.1016/j.jmaa.2005.02.026.

[12]

M. K. Kwong, On Lyapunov's inequality for disfocality, J. Math. Anal. Appl., 83 (1981), 486-494. doi: 10.1016/0022-247X（81）90137-2.

[13]

C. Lee, C. Yeh, C. Hong and R. P. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett., 17 (2004), 847-853. doi: 10.1016/j.aml.2004.06.016.

[14]

A. M. Liapunov, Problème général de la stabilité du mouvement, Fac. Sci. Univ. Toulouse., 2 (1907), 203-407.

[15]

Z. Nehari, Some eigenvalue estimates, J. D'analyse Math., 7 (1959), 79-88. doi: 10.1007/BF02787681.

[16]

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

[17]

B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl., 195 (1995), 527-536. doi: 10.1006/jmaa.1995.1372.

[18]

J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-Laplacian, Abstr. Appl. Anal., 2004 (2004), 147-153. doi: 10.1155/S108533750431002X.

[19]

T. W. Reid, A matrix equation related to a non-oscillation criterion and Lyapunov stability, Quart. Appl. Math. Soc., 23 (1965), 83-87.

[20]

T. W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl., 32 (1970), 424-434. doi: 10.1016/0022-247X(70)90308-2.

[21]

B. Singh, Forced oscillations in general ordinary differential equations, Tamkang Math. J., 6 (1976), 7-14. doi: euclid.hmj/1206135207.

[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. doi: 10.1016/j.jde.2011.08.002.

[23]

A. Tiryaki, M. Ünal and D. Cakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl., 332 (2007), 497-511. doi: 10.1016/j.jmaa.2006.10.010.

[24]

X. Wang, Stability criteria for linear periodic Hamiltonian systems, J. Math. Anal. Appl., 367 (2010), 329-336. doi: 10.1016/j.jmaa.2010.01.027.

[25]

X. Yang, On inequalities of Lyapunov type, Appl. Math. Comput., 134 (2003), 293-300. doi: 10.1016/S0096-3003(01)00283-1.

[26]

X. Yang, On Liapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134 (2003), 307-317. doi: 10.1016/S0096-3003(01)00285-5.

show all references

References:
[1]

S. S. Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math. J., 12 (1983), 105-112. doi: /327/1/HMJ12-105.

[2]

S. S. Cheng, Lyapunov inequalities for differential and difference equations, Hokkaido Math. Fasc. Math., 23 (1991), 25-41.

[3]

D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368-373. doi: 10.1016/j.amc.2010.01.010.

[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), 670-677. doi: 10.1016/0022-247X（75）90117-1.

[5]

S. B. Eliason, A Lyapunov inequality for a certain second order nonlinear differential equation, J. London Math. Soc., 2 (1970), 461-466.

[6]

S. B. Eliason, Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math., 27 (1974), 180-199. doi: 10.1137/0127015.

[7]

G. Sh. Guseinov and B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl., 45 (2003), 1399-1416. doi: 10.1016/S0898-1221(03)00095-6.

[8]

G. Sh. Guseinov and A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems, J. Math. Anal. Appl., 335 (2007), 1195-1206. doi: 10.1016/j.jmaa.2007.01.095.

[9]

P. Hartman and A. Wintner, On an oscillation criterion of Lyapunov, Amer. J. Math., 73 (1951), 885-890. doi: jstor.org/stable/2372122.

[10]

H. Hochstadt, A new proof of a stability estimate of Lyapunov, Proc. Amer. Math. Soc., 14 (1963), 525-526. doi: 10.1090/S0002-9939-1963-0149019-6.

[11]

L. Q. Jiang and Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl., 310 (2005), 579-593. doi: 10.1016/j.jmaa.2005.02.026.

[12]

M. K. Kwong, On Lyapunov's inequality for disfocality, J. Math. Anal. Appl., 83 (1981), 486-494. doi: 10.1016/0022-247X（81）90137-2.

[13]

C. Lee, C. Yeh, C. Hong and R. P. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett., 17 (2004), 847-853. doi: 10.1016/j.aml.2004.06.016.

[14]

A. M. Liapunov, Problème général de la stabilité du mouvement, Fac. Sci. Univ. Toulouse., 2 (1907), 203-407.

[15]

Z. Nehari, Some eigenvalue estimates, J. D'analyse Math., 7 (1959), 79-88. doi: 10.1007/BF02787681.

[16]

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

[17]

B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl., 195 (1995), 527-536. doi: 10.1006/jmaa.1995.1372.

[18]

J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-Laplacian, Abstr. Appl. Anal., 2004 (2004), 147-153. doi: 10.1155/S108533750431002X.

[19]

T. W. Reid, A matrix equation related to a non-oscillation criterion and Lyapunov stability, Quart. Appl. Math. Soc., 23 (1965), 83-87.

[20]

T. W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl., 32 (1970), 424-434. doi: 10.1016/0022-247X(70)90308-2.

[21]

B. Singh, Forced oscillations in general ordinary differential equations, Tamkang Math. J., 6 (1976), 7-14. doi: euclid.hmj/1206135207.

[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. doi: 10.1016/j.jde.2011.08.002.

[23]

A. Tiryaki, M. Ünal and D. Cakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl., 332 (2007), 497-511. doi: 10.1016/j.jmaa.2006.10.010.

[24]

X. Wang, Stability criteria for linear periodic Hamiltonian systems, J. Math. Anal. Appl., 367 (2010), 329-336. doi: 10.1016/j.jmaa.2010.01.027.

[25]

X. Yang, On inequalities of Lyapunov type, Appl. Math. Comput., 134 (2003), 293-300. doi: 10.1016/S0096-3003(01)00283-1.

[26]

X. Yang, On Liapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134 (2003), 307-317. doi: 10.1016/S0096-3003(01)00285-5.

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