Lyapunov-Type Inequalities for Two Classes of Difference Systems with Dirichlet Boundary Conditions

Year 2015, Volume: 28 Issue: 3, 473 - 482, 05.10.2015

Mustafa Aktaş , Devrim Çakmak , Merve Arıtürk

Abstract

In this paper, we establish Lyapunov-type inequalities for two classes of difference systems which improve all existing ones in the literature. Applying our inequalities, we obtain a lower bound for the eigenvalues of corresponding systems.

Keywords

Difference systems, Lyapunov-type inequalities, Dirichlet boundary conditions

References

• Aktaş, M. F., Çakmak, D., Tiryaki, A., “A note on Tang and He's paper”, Appl. Math. Comput. 218 (2012), 4867- 4871.
• Aktaş, M. F., “Lyapunov-type inequalities for a certain class of n-dimensional quasilinear systems”, Electron. J. Differential Equations 67 (2013), 1-8.
• Atkinson, F. V., “Discrete and Continuous Boundary Problems”, Academic Press, New York, NY, USA, 1964.
• Cheng, S. S., “A discrete analogue of the inequality of Lyapunov”, Hokkaido Math. J. 12 (1983), 105-112.
• Çakmak, D., Tiryaki, A., “On Lyapunov-type inequality for quasilinear systems”, Appl. Math. Comput. 216 (2010), 3584-3591.
• Çakmak, D., Tiryaki, A., “Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the (p₁,p₂,..., )-Laplacian”, J. Math. Anal. Appl. 369 (2010), 76-81.
• Liapunov, A. M., “Probleme general de la stabilite du mouvement”, Ann. Fac. Sci. Univ. Toulouse 2 (1907), 203- 407.
• Lin, S. H., Yang, G. S., “On discrete analogue of Lyapunov inequality”, Tamkang J. Math. 20 (1989), 169-186.
• Tang, X. H., He, X., “Lower bounds for generalized eigenvalues of the quasilinear systems”, J. Math. Anal. Appl. 385 (2012), 72-85.
• Tiryaki, A., Çakmak, D., Aktaş, M. F., “Lyapunov-type inequalities for a certain class of nonlinear systems”, Comput. Math. Appl. 64 (2012), 1804-1811.
• Tiryaki, A., Çakmak, D., Aktaş, M. F., “Lyapunov-type inequalities for two classes of Dirichlet quasilinear systems”, Math. Inequal. Appl. 17 (2014), 843-863.
• Ünal, M., Çakmak, D., Tiryaki, A., “A discrete analogue of Lyapunov-type inequalities for nonlinear systems”, Comput. Math. Appl. 55 (2008), 2631-2642.
• Wang, Y., Shi, Y., “Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions”, J. Math. Anal. Appl. 309 (2005), 56-69. 482
• GU J Sci, 28(3):473-482 (2015)/ Mustafa Fahri AKTAŞ, Devrim ÇAKMAK, Merve ARITÜRK
• Zhang, Q., Tang, X. H., “Lyapunov-type inequalities for quasilinear difference systems”, Discrete Dyn. Nat. Soc. 2012 (2012) Article ID 860598, 16 pages. [15] 25 (2012), 1830-1834.