Ulam-Hyers stability of fractional impulsive differential equations

Volume 11, Issue 8, pp 953--959 http://dx.doi.org/10.22436/jnsa.011.08.02

Publication Date: June 04, 2018 Submission Date: December 06, 2017 Revision Date: March 19, 2018 Accteptance Date: April 14, 2018

Download PDF

Download XML

2332 Downloads
4273 Views

Authors

Yali Ding - School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China.

Abstract

In this paper, we first prove the existence and uniqueness for a fractional differential equation with time delay and finite impulses on a compact interval. Secondly, Ulam-Hyers stability of the equation is established by Picard operator and abstract Gronwall's inequality.

Share and Cite

ISRP Style

Yali Ding, Ulam-Hyers stability of fractional impulsive differential equations, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 8, 953--959

AMA Style

Ding Yali, Ulam-Hyers stability of fractional impulsive differential equations. J. Nonlinear Sci. Appl. (2018); 11(8):953--959

Chicago/Turabian Style

Ding, Yali. "Ulam-Hyers stability of fractional impulsive differential equations." Journal of Nonlinear Sciences and Applications, 11, no. 8 (2018): 953--959

Keywords

Ulam-Hyers stability
fractional order impulsive equation
delay differential equation

MSC

26A33
45N05

References

[1] S. Abbas, M. Benchohra , Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput., 257 (2015), 190–198.

[2] N. Ali, B. B. Fatima, K. Shah, R. A. Khan, Hyers-Ulam stability of a class of nonlocal boundary value problem having triple solutions, Int. J. Appl. Comput. Math., 4 (2018), 12 pages.

[3] 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.

[4] S. András, A. R. Mészáros , Ulam-Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 219 (2013), 4853–4864.

[5] J. Brzdęk, N. Eghbali, On approximate solutions of some delayed fractional differential equations, Appl. Math. Lett., 54 (2016), 31–35.

[6] Z. Gao, X. Yu, J. Wang, Exp-type Ulam-Hyers stability of fractional differential equations with positive constant coefficient, Adv. Difference Equ., 2015 (2015), 10 pages.

[7] I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305–320.

[8] K. Shah, H. Khalil, R. A. Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Chaos Solitons Fractals, 77 (2015), 240–246.

[9] J.-R. Wang, M. Fečkan, Y. Zhou , Ulam’s type stability of impulsive ordinary differential equations , J. Math. Anal. Appl., 395 (2012), 258–264.

[10] J.-R. Wang, X. Li , Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258 (2015), 72–83.

[11] J.-R. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 10 pages.

[12] J.-R. Wang, K. Shah, A. Ali, Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods Appl. Sci., 41 (2018), 2392–2402.

[13] C. Wang, T.-Z. Xu, , Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives, Appl. Math., 60 (2015), 383–393.

[14] J.-R. Wang, Y. Zhang, Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 63 (2014), 1181–1190.

[15] J.-R. Wang, Y. Zhou, Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649–657.

[16] X.-J. Yang, C.-D. Li, T.-W. Huang, Q.-K. Song , Mittag-Leffler stability analysis of nonlinear fractional-order systems with impulses , Appl. Math. Comput., 293 (2017), 416–422.

[17] A. Zada, W. Ali, S. Farina , Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Methods Appl. Sci., 40 (2017), 5502–5514.

[18] A. Zada, S. Faisal, Y.-J. Li , On the Hyers-Ulam stability of first-order impulsive delay differential equations, J. Funct. Spaces, 2016 (2016), 6 pages.