Lyapunov-type inequalities for certain higher order fractional differential equations
-
1847
Downloads
-
3483
Views
Authors
Xuhuan Wang
- Department of Education Science, Pingxiang University, Pingxiang, Jiangxi 337055, China.
Youhua Peng
- Department of Mathematics, Pingxiang University, Pingxiang, Jiangxi 337055, China.
Wanchun Lu
- Department of Mathematics, Pingxiang University, Pingxiang, Jiangxi 337055, China.
Abstract
This paper generalizes the well-known Lyapunov-type inequality for certain higher
order fractional differential equations. The investigation is based on a construction of Green's functions and finding its corresponding maximum value. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.
Share and Cite
ISRP Style
Xuhuan Wang, Youhua Peng, Wanchun Lu, Lyapunov-type inequalities for certain higher order fractional differential equations, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 5064--5071
AMA Style
Wang Xuhuan, Peng Youhua, Lu Wanchun, Lyapunov-type inequalities for certain higher order fractional differential equations. J. Nonlinear Sci. Appl. (2017); 10(9):5064--5071
Chicago/Turabian Style
Wang, Xuhuan, Peng, Youhua, Lu, Wanchun. "Lyapunov-type inequalities for certain higher order fractional differential equations." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 5064--5071
Keywords
- Fractional differential equations
- Lyapunov-type inequality
- Green's function
- boundary value problem
MSC
References
-
[1]
N. Al Arifi, I. Altun, M. Jleli, A. Lashin, B. Samet, Lyapunov-type inequalities for a fractional p-Laplacian equation, J. Inequal. Appl., 2016 (2016), 11 pages.
-
[2]
A. Ali, B. Samet, K. Shah, R. A. Khan , Existence and stability of solution to a toppled systems of differential equations of non-integer order, Bound. Value Probl., 2017 (2017), 13 Pages.
-
[3]
A. Ali, K. Shah, R. A. Khan, Existence of positive solutions to a coupled system of nonlinear fractional order differential equations with m-point boundary conditions, Bull. Math. Anal. Appl., 8 (2016), 1–11.
-
[4]
I. Cabrera, B. Lopez, K. Sadarangan , Lyapunov type inequalities for a fractional two-point boundary value problem, Math. Methods Appl. Sci., 40 (2017), 3409–3414.
-
[5]
I. Cabrera, K. Sadarangani, B. Samet, Hartman-Wintner-type inequalities for a class of nonlocal fractional boundary value problems, Math. Methods Appl. Sci., 40 (2017), 129–136.
-
[6]
D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368–373.
-
[7]
R. A. C. Ferreira, A Lyapunov-type inequality for a fractional boundary value problem, Fract. Calc. Appl. Anal., 16 (2013), 978–984.
-
[8]
R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058–1063.
-
[9]
R. A. C. Ferreira, Some discrete fractioal Lyapunov-type inequalities, Fract. Differ. Calc., 5 (2015), 87–92.
-
[10]
K. Ghanbari, Y. Gholami, Lyapunov type inequalities for fractional Sturm-Liouville problems and fractional hamiltonian systems and applications, J. Fract. Calc. Appl., 7 (2016), 176–188.
-
[11]
M. Jleli, R. Lakhdar, B. Samet, A Lyapunov-type inequality for a fractional differential equation under a Robin boundary condition, J. Funct. Spaces., 2015 (2015), 5 pages.
-
[12]
M. Jleli, J. J. Nieto, B. Samet , Lyapunov-type inequalities for a higher order fractional differential equation with fractional integral boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 17 pages.
-
[13]
M. Jleli, B. Samet , Lyapunov-type inequalities for a Fractional Differential Equation with mixed boundary conditions, Math. Inequal. Appl., 18 (2015), 443–451.
-
[14]
M. Jleli, B. Samet , Lyapunov-type inequalities for fractional boundary-value problems, Electron. J. Differential Equations, 2015 (2015), 11 pages.
-
[15]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)
-
[16]
A. M. Liapunov, Probleme général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1907), 203–474.
-
[17]
D. O0Regan, B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), 10 pages.
-
[18]
B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl., 195 (1995), 527–536.
-
[19]
N. Parhi, S. Panigrahi, On Liapunov-Type Inequality for Third-Order Differential Equations, J. Math. Anal. Appl., 233 (1999), 445–460.
-
[20]
N. Parhi, S. Panigrahi, Lyapunov-type inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31–46.
-
[21]
J. Rong, C. Z. Bai, Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions, Adv. Difference Equ., 2015 (2015), 10 pages.
-
[22]
K. Shah, H. Khalil, R. A. Khan , Upper and lower solutions to a coupled system of nonlinear fractional differential equations, Progr. Fract. Differ. Appl., 2 (2016), 31–39.
-
[23]
S. Sitho, S. K. Ntouyas, W. Yukunthorn, J. Tariboon, Lyapunov’s type inequalities for hybrid fractional differential equations, J. Inequal. Appl., 2016 (2016), 13 pages.
-
[24]
A. Tiryaki , Recent developments of Lyapunov-type inequalities, Adv. Dyn. Syst. Appl., 5 (2010), 231–248.
-
[25]
X.-H. Wang, L. Lu, J.-T. Liang, Multiple solutions of nonlinear fractional impulsive integro-differential equations with nonlinear boundary conditions, Math. Slovaca, 66 (2016), 1105–1114.
-
[26]
X.-J. Yang, On Lyapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134 (2003), 307–317.
-
[27]
X.-J. Yang, Y.-I. Kim, K. Lo, Lyapunov-type inequality for a class of odd-order differential equations, J. Comput. Appl. Math., 234 (2010), 2962–2968.
-
[28]
X.-J. Yang, K. Lo, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884–3890.
-
[29]
Q.-M. Zhang, X. H. Tang, Lyapunov-type inequalities for even order difference equations, Appl. Math. Lett., 25 (2012), 1830–1834.