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Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains
Lyapunov-type inequalities for even order differential equations
1. | School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China |
$x^{(2n)}(t)+(-1)^{n-1}q(t)x(t)=0, $
which are sharper than all related existing ones.
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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