Ulam-Hyers stability of fractional impulsive differential equations
- School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China.
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.
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Yali Ding, Ulam-Hyers stability of fractional impulsive differential equations, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 8, 953--959
Ding Yali, Ulam-Hyers stability of fractional impulsive differential equations. J. Nonlinear Sci. Appl. (2018); 11(8):953--959
Ding, Yali. "Ulam-Hyers stability of fractional impulsive differential equations." Journal of Nonlinear Sciences and Applications, 11, no. 8 (2018): 953--959
- Ulam-Hyers stability
- fractional order impulsive equation
- delay differential equation
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.
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.
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.
S. András, A. R. Mészáros , Ulam-Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 219 (2013), 4853–4864.
J. Brzdęk, N. Eghbali, On approximate solutions of some delayed fractional differential equations, Appl. Math. Lett., 54 (2016), 31–35.
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.
I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305–320.
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.
J.-R. Wang, M. Fečkan, Y. Zhou , Ulam’s type stability of impulsive ordinary differential equations , J. Math. Anal. Appl., 395 (2012), 258–264.
J.-R. Wang, X. Li , Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258 (2015), 72–83.
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.
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.
C. Wang, T.-Z. Xu, , Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives, Appl. Math., 60 (2015), 383–393.
J.-R. Wang, Y. Zhang, Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 63 (2014), 1181–1190.
J.-R. Wang, Y. Zhou, Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649–657.
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.
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.
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.