Abstract
We revisit motivation of the fractional difference equations and some recent applications to image encryption. Then stability of impulsive fractional difference equations is investigated in this paper. The fractional sum equation is considered and impulsive effects are introduced into discrete fractional calculus. A class of impulsive fractional difference equations are proposed. A discrete comparison principle is given and asymptotic stability of nonlinear fractional difference equation are discussed. Finally, an impulsive Mittag–Leffler stability is defined. The numerical result is provided to support the analysis.
Acknowledgments
This study was financially supported by Sichuan Science and Technology Support Program (Grant No. 2018JY0120) and China Postdoctoral Science Foundation (Grant No. 2016M602632).
We thank Prof. Fu-Lai Chen and Dr. Chao Wang for helpful discussion and suggestion, as well as to the referees for interesting comments and remarks.
References
[1] T. Abdeljawad, On Riemann and Caputo fractional differences. Comput. Math. Appl. 62, No 3 (2011), 1602–1611.10.1016/j.camwa.2011.03.036Search in Google Scholar
[2] R. Abu–Saris, Q. Al–Mdallal, On the asymptotic stability of linear system of fractional–order difference equations. Fract. Calc. Appl. Anal. 16, No 3 (2013), 613–629; 10.2478/s13540-013-0039-2; https://www.degruyter.com/view/j/fca.2013.16.issue-3/issue-files/fca.2013.16.issue-3.xml.Search in Google Scholar
[3] R. Agarwal, S. Hristova, D. O’Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19, No 2 (2016), 290–318; 10.1515/fca-2016-0017; https://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml.Search in Google Scholar
[4] R. Agarwal, S. Hristova, D. O’Regan, Stability of solutions to impulsive Caputo fractional differential equations. Elect. J. Differ. Equat. 2016, No 58 (2016), 1–12.10.1163/15685365-12341515Search in Google Scholar
[5] G.A. Anastassiou, About discrete fractional calculus with inequalities, In: Intelligent Mathematics: Computational Analysis, Springer, 575–585 (2011).10.1007/978-3-642-17098-0_35Search in Google Scholar
[6] F.M. Atici, P.W. Eloe, A transformmethod in discrete fractional calculus. Int. J. Differ. Equa. 2, No 2 (2007), 165–176.Search in Google Scholar
[7] F.M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus. Proc. Amer. Math. Soc. 137, No 3 (2007), 981–989.10.1090/S0002-9939-08-09626-3Search in Google Scholar
[8] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods. World Scientific Publ., Singapore (2012).10.1142/8180Search in Google Scholar
[9] D. Baleanu, G.C. Wu, Y.R. Bai, F.L. Chen, Stability analysis of Caputo–like discrete fractional systems. Commun. Nonlinear Sci. Numer. Simulat. 48 (2017), 520–530.10.1016/j.cnsns.2017.01.002Search in Google Scholar
[10] N.R.O. Bastos, R.A.C. Ferreira, D.F.M. Torres, Discrete–time fractional variational problems. Sign. Proc. 91, No 3 (2011), 513–524.10.1016/j.sigpro.2010.05.001Search in Google Scholar
[11] J. Cermak, I. Gyori, L. Nechvatal, On explicit stability conditions for a linear fractional difference system. Fract. Calc. Appl. Anal. 18, No 3 (2015), 651–672; 10.1515/fca-2015-0040; https://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.Search in Google Scholar
[12] S.K. Choi, N. Koo, A note on linear impulsive fractional differential equations. J. Chungcheong Math. Soc. 28, No 4 (2015), 583–590.10.14403/jcms.2015.28.4.583Search in Google Scholar
[13] C.S. Drapaca, Fractional calculus in neuronal electro mechanics. J. Mech. Mater. Struct. 12, No 1 (2017), 35–55.10.2140/jomms.2017.12.35Search in Google Scholar
[14] E. Edelman, V.E. Tarasov, Fractional standard map. Phys. Let. A374, No 2 (2009), 279–285.10.1016/j.physleta.2009.11.008Search in Google Scholar
[15] M. Feckan, Y. Zhou, J.R. Wang, On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat. 17, No 7 (2012), 3050–3060.10.1016/j.cnsns.2011.11.017Search in Google Scholar
[16] C.S. Goodrich, A.C. Peterson, Discrete Fractional Calculus. Springer, New York (2016).10.1007/978-3-319-25562-0Search in Google Scholar
[17] L.L. Huang, D. Baleanu, G.C. Wu, A new application of the fractional logistic map. Rom. J. Phys. 61, No 7 (2016), 1172–1179.Search in Google Scholar
[18] M.T. Holm, The theory of discrete fractional calculus: Development and application. Ph.D. Dissertation, University of Nebraska–Lincoln (2011).Search in Google Scholar
[19] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006).Search in Google Scholar
[20] V.S. Kiryakova, Generalized Fractional Calculus and Applications. Longman Sci. & Techn., Harlow and J. Wiley. & Sons Inc, New York (1994).Search in Google Scholar
[21] N. Laskin, Fractional quantum mechanics and Levy path integrals. Phys. Lett. A268, No 4 (1999), 298–305.10.1016/S0375-9601(00)00201-2Search in Google Scholar
[22] Y. Li, Y.Q. Chen, I. Podlubny, Stability of fractional–order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput. Math. Appl. 59, No 5 (2010), 1810–1821.10.1016/j.camwa.2009.08.019Search in Google Scholar
[23] J.A.T. Machado, A.M. Lopes, Relative fractional dynamics of stock market. Nonlinear Dyn. 86, No 3 (2016), 1613–1619.10.1007/s11071-016-2980-1Search in Google Scholar
[24] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, No 1 (2000), 1–77.10.1016/S0370-1573(00)00070-3Search in Google Scholar
[25] K.S. Miller, B. Ross, Fractional difference calculus. In: Proc. of the Internat. Symp. on Univalent Functions, Fractional Calculus, and Their Applications, 139–152, Nihon University, Koriyama, Japan (1989).Search in Google Scholar
[26] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar
[27] C.M.A. Pinto, A.R.M. Carvalho, The role of synaptic transmission in a HIV model with memory. Appl. Math. Comput. 292 (2017), 76–95.10.1016/j.amc.2016.07.031Search in Google Scholar
[28] H.G. Sun, X.T. Liu, Y. Zhou, et. al., A fast semi–discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion equation, J. Comput. Phys. 345 (2017), 74–90.10.1016/j.jcp.2017.05.012Search in Google Scholar
[29] V.E. Tarasov, G.M. Zaslavsky, Fractional equations of kicked systems and discrete maps. J. Phys. A: Math. Theor. 41, No 43 (2008), Art. # 435101.10.1088/1751-8113/41/43/435101Search in Google Scholar
[30] J.R. Wang, M. Feckan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equa. 8, No 4 (2011), 345–361.10.4310/DPDE.2011.v8.n4.a3Search in Google Scholar
[31] J.R. Wang, M. Feckan, Y. Zhou, A survey of impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19, No 4 (2016), 806–831; 10.1515/fca-2016-0044; https://www.degruyter.com/view/j/fca.2016.19.issue-4/issue-files/fca.2016.19.issue-4.xml.Search in Google Scholar
[32] G.C. Wu, D. Baleanu, Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, No 1–2 (2014), 283–287.10.1007/s11071-013-1065-7Search in Google Scholar
[33] G.C. Wu, D. Baleanu, Z.X. Lin, Image encryption technique based on fractional chaotic time series. J. Vibr. Contr. 22, No 8 (2016), 2092–2099.10.1177/1077546315574649Search in Google Scholar
[34] G.C. Wu, D. Baleanu, S.D. Zeng, Several fractional differences and their applications to discrete maps. J. Appl. Nonlinear Dyn. 4, No 4 (2015), 339–348.10.5890/JAND.2015.11.001Search in Google Scholar
[35] G.C. Wu, D. Baleanu, W.H. Luo, Lyapunov functions for Riemann–Liouville–like discrete fractional equations. Appl. Math. Comput. 314 (2017), 228–236.Search in Google Scholar
[36] G.C. Wu, D. Baleanu, S.D. Zeng, Finite–time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Commun. Nonlinear Sci. Numer. Simulat. 57 (2018), 299–308.10.1016/j.cnsns.2017.09.001Search in Google Scholar
[37] G.C. Wu, L.G. Zeng, D. Baleanu, X.C. Shi, F. Wu, One method for generating chaotic series by use of discrete fractional maps. Data base: Chinese Patent No. 2014100338357[P], 2014-1-24.Search in Google Scholar
[38] H. Xiao, Y. Ma, C. Li, Chaotic vibration in fractional maps. J. Vibr. Contr. 20, No 7 (2014), 964–972.10.1177/1077546312473769Search in Google Scholar
© 2018 Diogenes Co., Sofia
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Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 21–2–2018)
- Research Paper
- Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
- Research Paper
- Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
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- Two point fractional boundary value problems with a fractional boundary condition
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- Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
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