Lyapunov-type inequalities for some quasilinear dynamic system involving the \((p_1, p_2, \dots, p_m)\)-Laplacian on time scales. (English) Zbl 1235.93187

He, Xiaofei; Zhang, Qi-Ming

J. Appl. Math. 2011, Article ID 418136, 10 p. (2011).

Summary: We establish several new Lyapunov-type inequalities for some quasilinear dynamic system involving the \((p_1, p_2, \dots, p_m)\)-Laplacian on an arbitrary time scale \(\mathbb T\), which generalize and improve some related existing results including the continuous and discrete cases.

MSC:

93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34A08	Fractional ordinary differential equations

References:

[1]	S. Hilger, Einßmakettenkalküul mit anwendung auf zentrumsmannigfaltigkeiten, Ph.D. thesis, Universitäat Wüurzburg, 1988. · Zbl 0695.34001
[2]	S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990. · Zbl 0722.39001
[3]	S. Hilger, “Differential and difference calculus-unified!,” vol. 30, no. 5, pp. 2683-2694. · Zbl 0927.39002
[4]	B. Kaymakcalan, V. Lakshmikantham, and S. Sivasundaram, Dynamic Systems on Measure Chains, vol. 370 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996. · Zbl 0869.34039
[5]	M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Boston, Mass, USA, 2003. · Zbl 1056.34049
[6]	C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian systems: Difference Equations, Continued Fractions, and Riccati Equations, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic, Boston, Mass, USA, 1996. · Zbl 0860.39001
[7]	A. M. Lyapunov, “Problème général de la stabilité du mouvement,” Annde la Faculté, vol. 2, no. 9, pp. 203-474, 1907. · JFM 38.0738.07
[8]	Q. Zhang and X. H. Tang, “Lyapunov inequalities and stability for discrete linear Hamiltonian system,” Applied Mathematics and Computation, vol. 218, pp. 574-582, 2011. · Zbl 1229.39026
[9]	S. B. Eliason, “A Lyapunov inequality for a certain second order nonlinear differential equation,” Journal of the London Mathematical Society, vol. 2, pp. 461-466, 1970. · Zbl 0201.11802
[10]	B. G. Pachpatte, “Inequalities related to the zeros of solutions of certain second order differential equations,” Facta Universitatis. Series: Mathematics and Informatics, no. 16, pp. 35-44, 2001. · Zbl 1049.34039
[11]	X. He and Q. Zhang, “A discrete analogue of Lyapunov-type inequalities for nonlinear difference systems,” Computers & Mathematics with Applications, vol. 62, pp. 677-684, 2011. · Zbl 1298.39005
[12]	S. B. Eliason, “Lyapunov type inequalities for certain second order functional differential equations,” SIAM Journal on Applied Mathematics, vol. 27, pp. 180-199, 1974. · Zbl 0292.34077
[13]	R. S. Dahiya and B. Singh, “A Lyapunov inequality and nonoscillation theorem for a second order non-linear differential-difference equation,” vol. 7, pp. 163-170, 1973. · Zbl 0239.34033
[14]	B. G. Pachpatte, “On Lyapunov-type inequalities for certain higher order differential equations,” Journal of Mathematical Analysis and Applications, vol. 195, no. 2, pp. 527-536, 1995. · Zbl 0844.34014
[15]	X. Yang, “On Liapunov-type inequality for certain higher-order differential equations,” Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 307-317, 2003. · Zbl 1030.34019
[16]	X. Yang, “On inequalities of Lyapunov type,” Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 293-300, 2003. · Zbl 1030.34018
[17]	X. Yang and K. Lo, “Lyapunov-type inequality for a class of even-order differential equations,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3884-3890, 2010. · Zbl 1202.34021
[18]	D. Cakmak and A. Tiryaki, “On Lyapunov-type inequality for quasilinear systems,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3584-3591, 2010. · Zbl 1208.34022
[19]	D. Cakmak and A. Tiryaki, “Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the p1,p2,\cdots ,pn-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 76-81, 2010. · Zbl 1216.34019
[20]	C. Lee, C. Yeh, C. Hong, and R. P. Agarwal, “Lyapunov and Wirtinger inequalities,” Applied Mathematics Letters, vol. 17, no. 7, pp. 847-853, 2004. · Zbl 1062.34005
[21]	J. P. Pinasco, “Lower bounds for eigenvalues of the one-dimensional p-Laplacian,” Abstract and Applied Analysis, vol. 2004, no. 2, pp. 147-153, 2004. · Zbl 1074.34080
[22]	R. P. Agarwal, M. Bohner, and P. Rehak, “Half-linear dynamic equations,” Nonlinear Analysis and Applications, vol. 1, pp. 1-56, 2003. · Zbl 1056.34049
[23]	L. Q. Jiang and Z. Zhou, “Lyapunov inequality for linear Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 310, no. 2, pp. 579-593, 2005. · Zbl 1076.37053
[24]	Z. He, “Existence of two solutions of m-point boundary value problem for second order dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 97-109, 2004. · Zbl 1149.34310
[25]	X. He, Q. Zhang, and X. H. Tang, “On inequalities of Lyapunov for linear Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 381, pp. 695-705, 2011. · Zbl 1229.34137
[26]	S. H. Saker, “Oscillation of nonlinear dynamic equations on time scales,” Applied Mathematics and Computation, vol. 148, no. 1, pp. 81-91, 2004. · Zbl 1075.34028
[27]	M. Bohner, S. Clark, and J. Ridenhour, “Lyapunov inequalities for time scales,” Journal of Inequalities and Applications, vol. 7, no. 1, pp. 61-77, 2002. · Zbl 1088.34503
[28]	M. Ünal and D. Cakmak, “Lyapunov-type inequalities for certain nonlinear systems on time scales,” Turkish Journal of Mathematics, vol. 32, no. 3, pp. 255-275, 2008. · Zbl 1166.34005
[29]	X. H. Tang and X. He, “Lower bounds for generalized eigenvalues of the quasilinear systems,” Journal of Mathematical Analysis and Applications, vol. 385, pp. 72-85, 2012. · Zbl 1247.34129