International Journal of Differential Equations

Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications

Lakshman Mahto, Syed Abbas, and Angelo Favini

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We use Sadovskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order α(0,1) with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.

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Int. J. Differ. Equ., Volume 2013, Special Issue (2012), Article ID 704547, 11 pages.

Received: 6 May 2012
Accepted: 21 November 2012
First available in Project Euclid: 24 January 2017

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Mahto, Lakshman; Abbas, Syed; Favini, Angelo. Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications. Int. J. Differ. Equ. 2013, Special Issue (2012), Article ID 704547, 11 pages. doi:10.1155/2013/704547.

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  • V. D. Milman and A. D. Myškis, “On the stability of motion in the presence of impulses,” Siberial Mathematical Journal, vol. 1, pp. 233–237, 1960.
  • A. M. Samoilenko and N. A. Perestyuk, Differential Equations With Impulses, Viska Scola, Kiev, Ukraine, 1987.
  • V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
  • D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effects, Horwood, Chichister, UK, 1989.
  • D. D. Bainov and P. S. Simeonov, Impulsive DiffErential Equations: Periodic Solutions and Its Applications, Longman Scientific and Technical Group, Harlow, UK, 1993.
  • D. Bainov and V. Covachev, Impulsive Differential Equations with a Small Parameter, vol. 24 of Series on Advances in Mathematics for Applied Sciences, World Scientific, River Edge, NJ, USA, 1994.
  • M. Benchohra, J. Henderson, and S. K. Ntonyas, Impulsive Diffrential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, NY, USA, 2006.
  • J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977.
  • F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, pp. 291–348, Springer, Vienna, Austria, 1997.
  • K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, North-Holland Mathematics Studies, 2006.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in Fractional Calculus, Springer, Dordrecht, The Netherlands, 2007.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
  • S. Abbas, “Existence of solutions to fractional order ordinary and delay differential equations and applications,” Electronic Journal of Differential Equations, vol. 2011, no. 9, pp. 1–11, 2011.
  • S. Abbas, M. Banerjee, and S. Momani, “Dynamical analysis of fractional-order modified logistic model,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1098–1104, 2011.
  • E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 542–553, 2007.
  • S. B. Hadid, “Local and global existence theorems on differential equations of non-integer order,” Journal of Fractional Calculus, vol. 7, pp. 101–105, 1995.
  • R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1–10, 2007.
  • V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.
  • R. P. Agarwal, M. Benchohra, and B. A. Slimani, “Existence results for differential equations with fractional order and impulses,” Georgian Academy of Sciences. A. Razmadze Mathematical Institute. Memoirs on Differential Equations and Mathematical Physics, vol. 44, pp. 1–21, 2008.
  • M. Benchohra and B. A. Slimani, “Existence and uniqueness of solutions to impulsive fractional differential equations,” Electronic Journal of Differential Equations, vol. 2009, no. 10, pp. 1–11, 2009.
  • M. Fečkan, Y. Zhou, and J. Wang, “On the concept and existence of solution for impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050–3060, 2012.
  • D. Xu, Y. Hueng, and L. Ling, “Existence of positive solutions of an Impulsive Delay Fishing model,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 2, pp. 89–94, 2011.
  • C. Giannantoni, “The problem of the initial conditions and their physical meaning in linear differential equations of fractional order,” Applied Mathematics and Computation, vol. 141, no. 1, pp. 87–102, 2003.
  • N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5, pp. 765–771, 2006.
  • I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002, Dedicated to the 60th anniversary of Prof. Francesco Mainard.
  • E. Zeidler, Non-Linear Functional Analysis and Its Application: Fixed Point-Theorems, vol. 1, Springer, New York, NY, USA, 1986.
  • B. N. Sadovskiĭ, “On a fixed point principle,” Functional Analysis and Its Applications, vol. 1, no. 2, pp. 74–76, 1967.
  • H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007.