[1] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., U.S.A.
27(1941), 222-224.
[2] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int.
J. Math., 23, (5), (2012), 9 pp.
[3] R. W. Ibrahim, Ulam stability for fractional differential equation in complex domain, Abstr.
Appl. Anal., 2012, (2012), 1-8.
[4] R. W. Ibrahim, Ulam-Hyers stability for Cauchy fractional differential equation in the unit
disk, Abstr. Appl. Anal., 2012, (2012), 1-10.
[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] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equa-
tions, John wiley, New York, 1993.
[7] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[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] Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math.
Soc. 72, (1978), 297-300.
[10] S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science eds., Wiley, New York,
1960.
[11] J. Wang and X. Li, Ulam-Hyers stability of fractional Langevin equations, Appl. Math.
Comput., 258, (2015), 72-83.
[12] J. Wang and Z. Lin, Ulam's type stability of Hadamard type fractional integral equations,
Filomat, 28 (7), (2014), 1323-1331.
[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] 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] 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] 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] 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] 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] W. Wei, Xuezhu. Li and Xia Li, New stability results for fractional integral equation, Com-
put. Math. Appl., 64 (10), (2012), 3468-3476.
[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.
Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk
Year 2019,
Volume: 7 Issue: 2, 264 - 267, 15.10.2019
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.
[1] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., U.S.A.
27(1941), 222-224.
[2] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int.
J. Math., 23, (5), (2012), 9 pp.
[3] R. W. Ibrahim, Ulam stability for fractional differential equation in complex domain, Abstr.
Appl. Anal., 2012, (2012), 1-8.
[4] R. W. Ibrahim, Ulam-Hyers stability for Cauchy fractional differential equation in the unit
disk, Abstr. Appl. Anal., 2012, (2012), 1-10.
[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] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equa-
tions, John wiley, New York, 1993.
[7] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[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] Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math.
Soc. 72, (1978), 297-300.
[10] S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science eds., Wiley, New York,
1960.
[11] J. Wang and X. Li, Ulam-Hyers stability of fractional Langevin equations, Appl. Math.
Comput., 258, (2015), 72-83.
[12] J. Wang and Z. Lin, Ulam's type stability of Hadamard type fractional integral equations,
Filomat, 28 (7), (2014), 1323-1331.
[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] 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] 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] 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] 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] 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] W. Wei, Xuezhu. Li and Xia Li, New stability results for fractional integral equation, Com-
put. Math. Appl., 64 (10), (2012), 3468-3476.
[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.
Eghbali, N., & Kalvandi, V. (2019). Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk. Konuralp Journal of Mathematics, 7(2), 264-267.
AMA
Eghbali N, Kalvandi V. Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk. Konuralp J. Math. October 2019;7(2):264-267.
Chicago
Eghbali, Nasrin, and Vida Kalvandi. “Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk”. Konuralp Journal of Mathematics 7, no. 2 (October 2019): 264-67.
EndNote
Eghbali N, Kalvandi V (October 1, 2019) Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk. Konuralp Journal of Mathematics 7 2 264–267.
IEEE
N. Eghbali and V. Kalvandi, “Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk”, Konuralp J. Math., vol. 7, no. 2, pp. 264–267, 2019.
ISNAD
Eghbali, Nasrin - Kalvandi, Vida. “Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk”. Konuralp Journal of Mathematics 7/2 (October 2019), 264-267.
JAMA
Eghbali N, Kalvandi V. Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk. Konuralp J. Math. 2019;7:264–267.
MLA
Eghbali, Nasrin and Vida Kalvandi. “Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk”. Konuralp Journal of Mathematics, vol. 7, no. 2, 2019, pp. 264-7.
Vancouver
Eghbali N, Kalvandi V. Mittag-Leffler-Hyers-Ulam Stability for Cauchy Fractional Differential Equation in the Unit Disk. Konuralp J. Math. 2019;7(2):264-7.