Lyapunov-type inequalities for third order linear differential equations with two points boundary conditions

Abstract

In this paper, by using Green's functions for second order differential equations, we establish new Lyapunov-type inequalities for third order linear differential equations with two points boundary conditions. By using such inequalities, we obtain sharp lower bounds for the eigenvalues of corresponding equations.

Keywords

• Lyapunov-type inequalities,Green's functions,two points boundary conditions

References

1. M.F. Aktas, Lyapunov-type inequalities for $n$-dimensional quasilinear systems, Electron. J. Differential Equations 67, 1-8, 2013.
2. M.F. Aktas, D. Çakmak and A. Tiryaki, A note on Tang and He’s paper, Appl. Math. Comput. 218, 4867-4871, 2012.
3. M.F. Aktas, D. Çakmak and A. Tiryaki, Lyapunov-type inequality for quasilinear systems with anti-periodic boundary conditions, J. Math. Inequal. 8, 313-320, 2014.
4. M.F. Aktas, D. Çakmak and A. Tiryaki, On the Lyapunov-type inequalities of a threepoint boundary value problem for third order linear differential equations, Appl. Math. Lett. 45, 1-6, 2015.
5. G. Borg, On a Liapounoff criterion of stability, Amer. J. Math. 71, 67-70, 1949.
6. A. Cabada, J.A. Cid and B. Maquez-Villamarin, Computation of Green’s functions for boundary value problems with Mathematica, Appl. Math. Comput. 219, 1919-1936, 2012.
7. D. Çakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput. 216, 368-373, 2010.
8. D. Çakmak, On Lyapunov-type inequality for a class of nonlinear systems, Math. Inequal. Appl. 16, 101-108, 2013.

9. D. Çakmak, M.F. Aktas and A. Tiryaki, Lyapunov-type inequalities for nonlinear systems involving the $(p_{1},p_{2},...,p_{n}) $-Laplacian, Electron. J. Differential Equations 128, 1-10, 2013.
10. D. Çakmak and A. Tiryaki, On Lyapunov-type inequality for quasilinear systems, Appl. Math. Comput. 216, 3584-3591, 2010.
11. D. Çakmak and A. Tiryaki, Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the $(p_{1},p_{2},...,p_{n}) $-Laplacian, J. Math. Anal. Appl. 369, 76-81, 2010.
12. P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964 an Birkhäuser, Boston 1982.
13. E.L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1926.
14. W.G. Kelley and A.C. Peterson, The Theory of Differential Equations, Classical and Qualitative, Springer, New York, 2010.
15. A. Liapunov, Probleme general de la stabilite du mouvement, Annales de la Faculte des Sciences de Toulouse pour les Sciences Mathematiques et les Sciences Physiques 2, 203-474, 1907.
16. A. Tiryaki, D. Çakmak and M.F. Aktas, Lyapunov-type inequalities for a certain class of nonlinear systems, Comput. Math. Appl. 64, 1804-1811, 2012.
17. A. Tiryaki, D. Çakmak and M.F. Aktas, Lyapunov-type inequalities for two classes of Dirichlet quasilinear systems, Math. Inequal. Appl. 17, 843-863, 2014.
18. A. Tiryaki, M. Ünal and D. Çakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl. 332, 497-511, 2007.
19. X. Yang, On inequalities of Lyapunov type, Appl. Math. Comput. 134, 293-300, 2003.