EXISTENCE, UNIQUENESS AND STABILITY OF RANDOM IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS

doi:10.1016/S0252-9602(16)30010-8

Abstract

Abstract:

In this article, we study the existence, uniqueness, stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional differential systems by relaxing the linear growth conditions. Finally, we give examples to illustrate its applications.

Key words: Fractional differential systems, random impulses, stability, Hyers-Ulam stability, Hyers-Ulam-Rassias stability

CLC Number:

34A08

A. Vinodkumar, K. Malar, M. Gowrisankar, P. Mohankumar. EXISTENCE, UNIQUENESS AND STABILITY OF RANDOM IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS[J].Acta mathematica scientia,Series B, 2016, 36(2): 428-442.

Trendmd

References

[1] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive Differential Equations. Singapore:World Scientific, 1989
[2] Samoilenko A M, Perestyuk N A. Impulsive Differential Equations. Singapore:World Scientific, 1995
[3] Iwankievicz R, Nielsen S R K. Dynamic response of non-linear systems to Poisson distributed random impulses. J Sound Vibration, 1992, 156:407-423
[4] Anguraj A, Vinodkumar A. Existence, uniqueness and stability results of random impulsive semilinear differential systems. Nonlinear Analysis:Hybrid Systems, 2010, 4:475-483
[5] Sanz-Serna J M, Stuart A M. Ergodicity of dissipative differential equations subject to random impulses. J Differential Equations, 1999, 155:262-284
[6] Tatsuyuki K, Takashi K, Satoshi S. Drift motion of granules in chara cells induced by random impulses due to the myosin-actin interaction. Physica A, 1998, 248:21-27
[7] Wu S J, Guo X L, Lin S Q. Existence and uniqueness of solutions to random impulsive differential systems. Acta Math Appl Sin, 2006, 22(4):595-600
[8] Wu S J, Guo X L, Zhou Y. p-moment stability of functional differential equations with random impulses. Comput Math Appl, 2006, 52:1683-1694
[9] Wu S J. The Euler scheme for random impulsive differential equations. Appl Math Comput, 2007, 191:164-175
[10] Anguraj A, Wu S, Vinodkumar A. Existence and exponential stability of semilinear functional differential equations with random impulses under non-uniqueness. Nonlinear Anal TMA, 2011, 74:331-342
[11] Gowrisankar M, Mohankumar P, Vinodkumar A. Stability results of random impulsive semilinear differential equations. Acta Mathematica Scientia, 2014, 34B(4):1055-1071
[12] Zhang S, Sun J. Stability analysis of second-order differential systems with Erlang distribution random impulses. Advances in Difference Equations, 2013, 2013(4):403-415
[13] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Elsevier Science BV, 2006
[14] Podlubny I. Fractional Differential Equations. San Diego:Academic Press, 1999
[15] Miller K S, Ross B. An Introduction to the Fractional Calculus and Differential Equations. New York:John Wiley, 1993
[16] Li, Y, Chen Y, Podlubny I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 2009, 45:1965-1969
[17] Li Y, Chen Y, Podlubny I. Stability of fractional-order nonlinear dynamic systems:Lyapunov direct method and generalized Mittag-Leffler stability. Comput Math Appl, 2010, 59:1810-1821
[18] Rus I A. Ulam stability of ordinary differential equations. Studia Univ Babe? Bolyai Math, 2009, 54:125-133
[19] Deng W. Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal, 2010, 72:1768-1777
[20] Wang J, Lv L, Zhou Y. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron J Qual Theory Differ Equ, 2011, 63:1-10
[21] Wang J, Lv L, Zhou Y. New concepts and results in stability of fractional differential equations. Commun Nonlinear Sci Numer Simul, 2012, 17:2530-2538
[22] Wang J, Zhou Y, Fe?kan M. Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput Math Appl, 2012, 64:3389-3405
[23] Michal Fe?kan, Yong Zhou, JinRongWang, On the concept and existence of solution for impulsive fractional differential equations. Commun Nonlinear Sci Numer Simulat, 2011, 17:3050-3060
[24] Li X, Wang J. Ulam-Hyers-Rassias stability of semilinear differential equations with impulses. Electron J Diff Equ, 2013, 2013(172):1-8
[25] Adda F B, Cresson J. Fractional differential equations and the schrödinger equation. Appl Math Comput, 2005, 161:323-345
[26] Agarwal R P, Benchohra M, Slimani B A. Existence results for differential equations with fractional order and impulses. Memoirs Differ Equ Math Phys, 2008, 44:1-21
[27] Benchohra M, Slimani B A. Existence and uniqueness of solutions to impulsive fractional differential equations. Electron J Differ Equ, 2009, 10:1-11
[28] Mophou G M. Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal, 2010, 72:1604-1605
[29] N' Guérékata G M. A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal, 2009, 70:1873-1876
[30] Ulam S M. A collection of Mathematical Problems. New York:Interscience Publishers, 1968
[31] Hyers D H. On the stability of the linear functional equation. Proc Nat Acad Sci, 1941, 27:222-224
[32] Rassias Th M. On the stability of linear mappings in Banach spaces. Proc Amer Soc, 1978, 72:297-300
[33] C?dariu L. Stabilitatea Ulam-Hyers-Bourgin pentru ecuatii functionale. Timisara:Ed Univ Vest Timisoara, 2007
[34] Li C P, Zhang F R. A survey on the stability of fractional differential equations. Eur Phys J Special Topics, 2011, 193:27-47
[35] Xu D Y, Wang X H, Yang Z G. Further results on existence-uniqueness for stochastic functional differential equation. Sci China Math, 2013, 56:1169-1180
[36] Xu Daoyi, Li Bing, Long Shujun, Teng Lingying, Moment estimate and existence for solutions of stochastic functional differential equations. Nonlinear Analysis, 2014, 108:128-143
[37] Daoyi Xu, Bing Li, Shujun Long, Lingying Teng. Corrigendum to "Moment estimate and existence for solutions of stochastic functional differential equations"[Nonlinear Anal:TMA 108(2014) 128-143]. Nonlinear Analysis, 2015, 114:40-41