American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant
  • CPAA Home
  • This Issue
  • Next Article
    Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains
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 & 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.  Google Scholar

[2]

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

[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.  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), 670-677. 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), 461-466.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Hartman and A. Wintner, On an oscillation criterion of Lyapunov, Amer. J. Math., 73 (1951), 885-890. 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), 525-526. 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), 579-593. 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), 486-494. 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), 847-853. 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), 203-407.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

T. W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl., 32 (1970), 424-434. 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), 7-14. 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), 497-511. 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), 329-336. doi: 10.1016/j.jmaa.2010.01.027.  Google Scholar

[25]

X. Yang, On inequalities of Lyapunov type, Appl. Math. Comput., 134 (2003), 293-300. 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), 307-317. 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), 105-112. doi: /327/1/HMJ12-105.  Google Scholar

[2]

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

[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.  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), 670-677. 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), 461-466.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Hartman and A. Wintner, On an oscillation criterion of Lyapunov, Amer. J. Math., 73 (1951), 885-890. 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), 525-526. 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), 579-593. 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), 486-494. 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), 847-853. 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), 203-407.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

T. W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl., 32 (1970), 424-434. 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), 7-14. 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), 497-511. 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), 329-336. doi: 10.1016/j.jmaa.2010.01.027.  Google Scholar

[25]

X. Yang, On inequalities of Lyapunov type, Appl. Math. Comput., 134 (2003), 293-300. 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), 307-317. doi: 10.1016/S0096-3003(01)00285-5.  Google Scholar

[1]

He Zhang, Xue Yang, Yong Li. Lyapunov-type inequalities and solvability of second-order ODEs across multi-resonance. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1133-1148. doi: 10.3934/dcdss.2017061
[2]

Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad, Saed F. Mallak, Hussam Alrabaiah. Lyapunov type inequality in the frame of generalized Caputo derivatives. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2335-2355. doi: 10.3934/dcdss.2020212
[3]

Ravi P. Agarwal, Abdullah Özbekler. Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2281-2300. doi: 10.3934/cpaa.2016037
[4]

Jun Zhou, Jun Shen, Weinian Zhang. A powered Gronwall-type inequality and applications to stochastic differential equations. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7207-7234. doi: 10.3934/dcds.2016114
[5]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
[6]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
[7]

Naoki Chigira, Nobuo Iiyori and Hiroyoshi Yamaki. Nonabelian Sylow subgroups of finite groups of even order. Electronic Research Announcements, 1998, 4: 88-90.

[8]

Naoki Fujino, Mitsuru Yamazaki. Burgers' type equation with vanishing higher order. Communications on Pure & Applied Analysis, 2007, 6 (2) : 505-520. doi: 10.3934/cpaa.2007.6.505
[9]

R.S. Dahiya, A. Zafer. Oscillation theorems of higher order neutral type differential equations. Conference Publications, 1998, 1998 (Special) : 203-219. doi: 10.3934/proc.1998.1998.203
[10]

Haibin Chen, Liqun Qi. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1263-1274. doi: 10.3934/jimo.2015.11.1263
[11]

Lixing Han. An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 583-599. doi: 10.3934/naco.2013.3.583
[12]

Juhong Kuang, Weiyi Chen, Zhiming Guo. Periodic solutions with prescribed minimal period for second order even Hamiltonian systems. Communications on Pure & Applied Analysis, 2022, 21 (1) : 47-59. doi: 10.3934/cpaa.2021166
[13]

James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137
[14]

To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694
[15]

Yuri V. Rogovchenko, Fatoş Tuncay. Interval oscillation of a second order nonlinear differential equation with a damping term. Conference Publications, 2007, 2007 (Special) : 883-891. doi: 10.3934/proc.2007.2007.883
[16]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225
[17]

Ruy Coimbra Charão, Juan Torres Espinoza, Ryo Ikehata. A second order fractional differential equation under effects of a super damping. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4433-4454. doi: 10.3934/cpaa.2020202
[18]

Hubert L. Bray, Marcus A. Khuri. A Jang equation approach to the Penrose inequality. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 741-766. doi: 10.3934/dcds.2010.27.741
[19]

Shu-Yu Hsu. Some results for the Perelman LYH-type inequality. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3535-3554. doi: 10.3934/dcds.2014.34.3535
[20]

Ioan Bucataru. A setting for higher order differential equation fields and higher order Lagrange and Finsler spaces. Journal of Geometric Mechanics, 2013, 5 (3) : 257-279. doi: 10.3934/jgm.2013.5.257

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy