Ulam-Hyers stability of fractional impulsive differential equations. (English) Zbl 1449.34274
Summary: 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.
MSC:
| 34K37 | Functional-differential equations with fractional derivatives |
| 34K45 | Functional-differential equations with impulses |
| 34K27 | Perturbations of functional-differential equations |
References:
| [1] | Abbas, S.; M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput., 257, 190-198 (2015) · Zbl 1338.35455 · doi:10.1016/j.amc.2014.06.073 |
| [2] | Ali, N.; Fatima, B. B.; Shah, K.; Khan, R. A., Hyers-Ulam stability of a class of nonlocal boundary value problem having triple solutions, Int. J. Appl. Comput. Math., 4, 1-12 (2018) · Zbl 1384.34009 · doi:10.1007/s40819-017-0447-9 |
| [3] | Ali, A.; Samet, B.; Shah, K.; Khan, R. A., Existence and stability of solution to a toppled systems of differential equations of non-integer order, Bound. Value Probl., 2017, 1-13 (2017) · Zbl 1361.34006 · doi:10.1186/s13661-017-0749-1 |
| [4] | András, S.; A. R. Mészáros, Ulam-Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 219, 4853-4864 (2013) · Zbl 1468.39015 · doi:10.1016/j.amc.2012.10.115 |
| [5] | Brzdęk, J.; Eghbali, N., On approximate solutions of some delayed fractional differential equations, Appl. Math. Lett., 54, 31-35 (2016) · Zbl 1381.34103 · doi:10.1016/j.aml.2015.10.004 |
| [6] | Gao, Z.; Yu, X.; Wang, J., Exp-type Ulam-Hyers stability of fractional differential equations with positive constant coefficient, Adv. Difference Equ., 2015, 1-10 (2015) · Zbl 1422.34032 · doi:10.1186/s13662-015-0578-4 |
| [7] | Rus, I. A., Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10, 305-320 (2009) · Zbl 1204.47071 |
| [8] | Shah, K.; Khalil, H.; Khan, R. A., Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Chaos Solitons Fractals, 77, 240-246 (2015) · Zbl 1353.34028 · doi:10.1016/j.chaos.2015.06.008 |
| [9] | Wang, J.-R.; Fečkan, M.; Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395, 258-264 (2012) · Zbl 1254.34022 · doi:10.1016/j.jmaa.2012.05.040 |
| [10] | Wang, J.-R.; X. Li, Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258, 72-83 (2015) · Zbl 1338.39047 · doi:10.1016/j.amc.2015.01.111 |
| [11] | Wang, J.-R.; Lv, L.; Zhou, Y., Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011, 1-10 (2011) · Zbl 1340.34034 · doi:10.14232/ejqtde.2011.1.63 |
| [12] | Wang, J.-R.; Shah, K.; Ali, A., Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods Appl. Sci., 41, 2392-2402 (2018) · Zbl 1390.34030 · doi:10.1002/mma.4748 |
| [13] | Wang, C.; Xu, T.-Z.; , Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives, Appl. Math., 60, 383-393 (2015) · Zbl 1363.34023 · doi:10.1007/s10492-015-0102-x |
| [14] | Wang, J.-R.; Zhang, Y., Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 63, 1181-1190 (2014) · Zbl 1296.34034 · doi:10.1080/02331934.2014.906597 |
| [15] | Wang, J.-R.; Zhou, Y.; Lin, Z., On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242, 649-657 (2014) · Zbl 1334.34022 · doi:10.1016/j.amc.2014.06.002 |
| [16] | Yang, X.-J.; Li, C.-D.; Huang, T.-W.; Q.-K. Song, Mittag-Leffler stability analysis of nonlinear fractional-order systems with impulses, Appl. Math. Comput., 293, 416-422 (2017) · Zbl 1411.34023 · doi:10.1016/j.amc.2016.08.039 |
| [17] | Zada, A.; Ali, W.; S. Farina, Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Methods Appl. Sci., 40, 5502-5514 (2017) · Zbl 1387.34026 · doi:10.1002/mma.4405 |
| [18] | Zada, A.; Faisal, S.; Y.-J. Li, On the Hyers-Ulam stability of first-order impulsive delay differential equations, J. Funct. Spaces, 2016, 1-6 (2016) · Zbl 1342.34100 · doi:10.1155/2016/8164978 |
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