Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk

Abstract

In this paper, we prove the Mittag-Leffler-Hyers-Ulam stability of Cauchy fractional differential equations in the unit disk for the linear and non-linear cases.

Keywords

• Mittag-Leffler-Hyers-Ulam Stability,Fractional calculus

References

1. [1] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., U.S.A. 27(1941), 222-224.
2. [2] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23, (5), (2012), 9 pp.
3. [3] R. W. Ibrahim, Ulam stability for fractional differential equation in complex domain, Abstr. Appl. Anal., 2012, (2012), 1-8.
4. [4] R. W. Ibrahim, Ulam-Hyers stability for Cauchy fractional differential equation in the unit disk, Abstr. Appl. Anal., 2012, (2012), 1-10.
5. [5] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Stydies, 204, Elsevier Science, B. V., Amsterdam, 2006.
6. [6] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equa- tions, John wiley, New York, 1993.
7. [7] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
8. [8] Sh. Peng and J. R.Wang, Existence and Ulam-Hyers stability of ODEs involving two Caputo fractional derivatives, Electronic Journal of Qualitative Theory of Differential Equations, 48-54 (52), (2015), 1-16.

9. [9] Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72, (1978), 297-300.
10. [10] S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science eds., Wiley, New York, 1960.
11. [11] J. Wang and X. Li, Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258, (2015), 72-83.
12. [12] J. Wang and Z. Lin, Ulam's type stability of Hadamard type fractional integral equations, Filomat, 28 (7), (2014), 1323-1331.
13. [13] J. Wang and Z. Lin, A class of impulsive nonautonomous differential equations and Ulam- Hyers-Rassias stability, Mathematical Methods in the Applied Sciences, 38 (5), (2015), 865-880.
14. [14] J. R.Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63, (2011), 1-10.
15. [15] J. R.Wang, L. Lv and Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat. 17, (2012), 2530-2538.
16. [16] J. R. Wang and Y. Zhang, Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization: A Journal of Mathematical Programming and opti- mization Research, 63 (8), (2014), 1181-1190.
17. [17] J. R.Wang, Y. Zhou and M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Appl. Math. Comput., 64, (2012), 3389-3405.
18. [18] J. R.Wang, Y. Zhou and Z. Lin, On a new class of impulsive fractional differential equations, App. Math. Comput., 242, (2014), 649-657.
19. [19] W. Wei, Xuezhu. Li and Xia Li, New stability results for fractional integral equation, Com- put. Math. Appl., 64 (10), (2012), 3468-3476.
20. [20] H. M. Srivastava, Y. Ling and G. Bao, Some distortion inequalities associated with the fractional derivatives of analytic and univalent functions, J .Ineq. Pure Appl. Math ., 2, (2001), 1-6.