×

Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary conditions. (English) Zbl 1428.34037

Ukr. Math. J. 70, No. 6, 837-850 (2018) and Ukr. Mat. Zh. 70, No. 6, 727-738 (2018).
At the beginning of the 20th century, Lyapunov proved a remarkable result for second-order linear differential equations with homogeneous Dirichlet boundary conditions known as Lyapunov’s inequality. Lyapunov’s inequality has found many applications in the study of various properties of solutions for ordinary differential equations, and as a consequence, has generated a lot of interest over the years.
There are many improvements and generalizations of Lyapunov’s inequality. Particularly, it has been generalized widely to the higher-order linear equations and systems.
The author of the present paper provides new Lyapunov-type inequalities for nonlinear cycled systems \[ \left(r_{i}(x)\phi_{p_{i}}(u_{i}')\right)'+f_{i}(x)\phi_{\alpha_{i}}(u_{i+1})=0 \] and nonlinear strongly coupled systems \[ \left(r_{1}(x)\phi_{p_{1}}(u_{i}')\right)'+f_{i}(x)\sum_{j=1}^{n}\phi_{p_{1}}(u_{j})=0 \] with homogeneous Dirichlet boundary conditions under some additional assumptions. The author’s results generalize and improve some known results.
Also, it is to note that the history of the subject and motivation of the investigation are well-represented. The obtained results continue studies of the subject and may be useful for researches in the field.
The list of references contains 35 items.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. F. Aktaş, “Lyapunov-type inequalities for <Emphasis Type=”Italic“>n-dimensional quasilinear systems,” Electron. J. Different. Equat., 1-8 (2013).
[2] M. F. Aktaş, D. Ҫakmak, and A. Tiryaki, “A note on Tang and He’s paper,” Appl. Math. Comput., 218, 4867-4871 (2012). · Zbl 1252.34098
[3] D. Ҫakmak, “Lyapunov-type integral inequalities for certain higher order differential equations,” Appl. Math. Comput., 216, 368-373 (2010). · Zbl 1189.26040
[4] D. Ҫakmak and A. Tiryaki, “Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the (<Emphasis Type=”Italic“>p1, <Emphasis Type=”Italic“>p2..., <Emphasis Type=”Italic“>p <Emphasis Type=”Italic“>n)- Laplacian,” J. Math. Anal. Appl., 369, 76-81 (2010). · Zbl 1216.34019 · doi:10.1016/j.jmaa.2010.02.043
[5] D. Ҫakmak and A. Tiryaki, “On Lyapunov-type inequality for quasilinear systems,” Appl. Math. Comput., 216, 3584-3591 (2010). · Zbl 1208.34022
[6] D. Ҫakmak, “On Lyapunov-type inequality for a class of nonlinear systems,” Math. Inequal. Appl., 16, 101-108 (2013). · Zbl 1262.34024
[7] K. M. Das and A. S. Vatsala, “Green’s function for <Emphasis Type=”Italic“>n-<Emphasis Type=”Italic“>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
[8] O. Dosly and P. Rehak, Half-Linear Differential Equations, North-Holland Mathematics Studies, 202, Elsevier, Amsterdam (2005). · Zbl 1090.34001
[9] S. B. Eliason, “Lyapunov type inequalities for certain second order functional differential equations,” SIAM J. Appl. Math., 27, 180-199 (1974). · Zbl 0292.34077 · doi:10.1137/0127015
[10] G. Guseinov and B. Kaymakçalan, “Lyapunov inequalities for discrete linear Hamiltonian system,” Comput. Math. Appl., 45, 1399-1416 (2003). · Zbl 1055.39029 · doi:10.1016/S0898-1221(03)00095-6
[11] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston (1982). · Zbl 0476.34002
[12] X. He and X. H. Tang, “Lyapunov-type inequalities for even-order differential equations,” Comm. Pure Appl. Anal., 11, 465-473 (2012). · Zbl 1276.34014 · doi:10.3934/cpaa.2012.11.465
[13] M. K. Kwong, “On Lyapunov’s inequality for disfocality,” J. Math. Anal. Appl., 83, 486-494 (1981). · Zbl 0504.34020 · doi:10.1016/0022-247X(81)90137-2
[14] E. K. Lee, Y. H. Lee, and I. Sim, “<Emphasis Type=”Italic“>C1-regularity of solutions for <Emphasis Type=”Italic“>p-Laplacian problems,” Appl. Math. Lett., 22, 759-765 (2009). · Zbl 1178.34025 · doi:10.1016/j.aml.2008.08.014
[15] C. Lee, C. Yeh, C. Hong, and R. P. Agarwal, “Lyapunov and Wirtinger inequalities,” Appl. Math. Lett., 17, 847-853 (2004). · Zbl 1062.34005 · doi:10.1016/j.aml.2004.06.016
[16] A. M. Liapunov, “Probleme general de la stabilite du mouvement,” Ann. Fac. Sci. Toulouse Math. (6), 2, 203-407 (1907).
[17] P. L. Napoli and J. P. Pinasco, “Estimates for eigenvalues of quasilinear elliptic systems,” J. Different. Equat., 227, 102-115 (2006). · Zbl 1100.35077 · doi:10.1016/j.jde.2006.01.004
[18] B. G. Pachpatte, “On Lyapunov-type inequalities for certain higher order differential equations,” J. Math. Anal. Appl., 195, 527-536 (1995). · Zbl 0844.34014 · doi:10.1006/jmaa.1995.1372
[19] N. Parhi and S. Panigrahi, “Liapunov-type inequality for higher order differential equations,” Math. Slovaca, 52, 31-46 (2002). · Zbl 1019.34039
[20] J. P. Pinasco, “Lower bounds for eigenvalues of the one-dimensional <Emphasis Type=”Italic“>p-Laplacian,” Abstr. Appl. Anal., No. 2, 147-153 (2004). · Zbl 1074.34080 · doi:10.1155/S108533750431002X
[21] J. P. Pinasco, “Comparison of eigenvalues for the <Emphasis Type=”Italic“>p-Laplacian with integral inequalities,” Appl. Math. Comput., 182, 1399-1404 (2006). · Zbl 1112.34069
[22] M. M. Rodrigues, “Lyapunov inequalities for nonlinear <Emphasis Type=”Italic“>p-Laplacian problems with weight functions,” Int. J. Math. Anal., 5, 1497-1506 (2011). · Zbl 1259.34016
[23] I. Sim and Y. H. Lee, “Lyapunov inequalities for one-dimensional <Emphasis Type=”Italic“>p-Laplacian problems with a singular weight function,” J. Inequal. Appl., Art. ID 865096, 9 p. (2010). · Zbl 1216.34023
[24] X. H. Tang and X. He, “Lower bounds for generalized eigenvalues of the quasilinear systems,” J. Math. Anal. Appl., 385, 72-85 (2012). · Zbl 1247.34129 · doi:10.1016/j.jmaa.2011.06.026
[25] A. Tiryaki, “Recent developments of Lyapunov-type inequalities,” Adv. Dynam. Syst. Appl., 5, 231-248 (2010).
[26] A. Tiryaki, D. Ҫakmak, and M. F. Aktaş, “Lyapunov-type inequalities for a certain class of nonlinear systems,” Comput. Math. Appl., 64, 1804-1811 (2012). · Zbl 1268.34063 · doi:10.1016/j.camwa.2012.02.019
[27] A. Tiryaki, M. Ünal, and D. Ҫakmak, “Lyapunov-type inequalities for nonlinear systems,” J. Math. Anal. Appl., 332, 497-511 (2007). · Zbl 1123.34037 · doi:10.1016/j.jmaa.2006.10.010
[28] M. Ünal, D. Ҫakmak, and A. Tiryaki, “A discrete analogue of Lyapunov-type inequalities for nonlinear systems,” Comput. Math. Appl., 55, 2631-2642 (2008). · Zbl 1142.39309 · doi:10.1016/j.camwa.2007.10.014
[29] M. Ünal and D. Ҫakmak, “Lyapunov-type inequalities for certain nonlinear systems on time scales,” Turkish J. Math., 32, 255-275 (2008). · Zbl 1166.34005
[30] X. Yang, “On Liapunov-type inequality for certain higher-order differential equations,” Appl. Math. Comput., 134, 307-317 (2003). · Zbl 1030.34019
[31] X. Yang and K. Lo, “Lyapunov-type inequality for a class of even-order differential equations,” Appl. Math. Comput., 215, 3884-3890 (2010). · Zbl 1202.34021
[32] X. Yang, Y. Kim, and K. Lo, “Lyapunov-type inequality for a class of odd-order differential equations,” J. Comput. Appl. Math., 234, 2962-2968 (2010). · Zbl 1198.26029 · doi:10.1016/j.cam.2010.04.008
[33] X. Yang, Y. Kim, and K. Lo, “Lyapunov-type inequality for a class of quasilinear systems,” Math. Comput. Model., 53, 1162-1166 (2011). · Zbl 1217.34022 · doi:10.1016/j.mcm.2010.11.083
[34] X. Wang, “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
[35] X. Wang, “Lyapunov-type inequalities for second-order half-linear differential equations,” J. Math. Anal. Appl., 382, 792-801 (2011). · Zbl 1236.34023 · doi:10.1016/j.jmaa.2011.04.075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.