On the Lyapunov theory for functional differential equations of fractional order

Stamova, Ivanka

doi:10.1090/proc/12822

On the Lyapunov theory for functional differential equations of fractional order
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by Ivanka M. Stamova PDF

Proc. Amer. Math. Soc. 144 (2016), 1581-1593 Request permission

Abstract:

In this paper efficient criteria for uniform asymptotic stability and boundedness of fractional-order functional differential equations are proved. To this end the Lyapunov-like functions and Mittag-Leffler functions are used.

References

M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008), no. 2, 1340–1350. MR 2386501, DOI 10.1016/j.jmaa.2007.06.021
Kai Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. An application-oriented exposition using differential operators of Caputo type. MR 2680847, DOI 10.1007/978-3-642-14574-2
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721
Johnny Henderson and Abdelghani Ouahab, Fractional functional differential inclusions with finite delay, Nonlinear Anal. 70 (2009), no. 5, 2091–2105. MR 2492146, DOI 10.1016/j.na.2008.02.111
R. Hilfer and H. J. Seybold, Computation of the generalized Mittag-Leffler function and its inverse in the complex plane, Integral Transforms Spec. Funct. 17 (2006), no. 9, 637–652. MR 2242716, DOI 10.1080/10652460600725341
Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. 69 (2008), no. 10, 3337–3343. MR 2450543, DOI 10.1016/j.na.2007.09.025
V. Lakshmikantham, S. Leela, and M. Sambandham, Lyapunov theory for fractional differential equations, Commun. Appl. Anal. 12 (2008), no. 4, 365–376. MR 2494983
Yan Li, YangQuan Chen, and Igor Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 (2009), no. 8, 1965–1969. MR 2879525, DOI 10.1016/j.automatica.2009.04.003
James H. Liu, Uniform asymptotic stability via Liapunov-Razumikhin technique, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2465–2471. MR 1257116, DOI 10.1090/S0002-9939-1995-1257116-8
Matilde Marcolli and Varghese Mathai, Towards the fractional quantum Hall effect: a noncommutative geometry perspective, Noncommutative geometry and number theory, Aspects Math., E37, Friedr. Vieweg, Wiesbaden, 2006, pp. 235–261. MR 2327308, DOI 10.1007/978-3-8348-0352-8_{1}2
Pham Huu Anh Ngoc, Novel criteria for exponential stability of functional differential equations, Proc. Amer. Math. Soc. 141 (2013), no. 9, 3083–3091. MR 3068962, DOI 10.1090/S0002-9939-2013-11554-6
Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR 1658022
Gani T. Stamov, Almost periodic solutions of impulsive differential equations, Lecture Notes in Mathematics, vol. 2047, Springer, Heidelberg, 2012. MR 2934087, DOI 10.1007/978-3-642-27546-3
Gani Tr. Stamov and Ivanka M. Stamova, Almost periodic solutions for impulsive fractional differential equations, Dyn. Syst. 29 (2014), no. 1, 119–132. MR 3170642, DOI 10.1080/14689367.2013.854737
Ivanka Stamova, Stability analysis of impulsive functional differential equations, De Gruyter Expositions in Mathematics, vol. 52, Walter de Gruyter GmbH & Co. KG, Berlin, 2009. MR 2604930, DOI 10.1515/9783110221824
Ivanka Stamova and Gani Stamov, Lipschitz stability criteria for functional differential systems of fractional order, J. Math. Phys. 54 (2013), no. 4, 043502, 11. MR 3088804, DOI 10.1063/1.4798234