Abstract and Applied Analysis

On Fractional Order Hybrid Differential Equations

Mohamed A. E. Herzallah and Dumitru Baleanu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0 < α < 1 . Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 389386, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412273272

Digital Object Identifier
doi:10.1155/2014/389386

Mathematical Reviews number (MathSciNet)
MR3191039

Zentralblatt MATH identifier
07022286

Citation

Herzallah, Mohamed A. E.; Baleanu, Dumitru. On Fractional Order Hybrid Differential Equations. Abstr. Appl. Anal. 2014 (2014), Article ID 389386, 7 pages. doi:10.1155/2014/389386. https://projecteuclid.org/euclid.aaa/1412273272


Export citation

References

  • D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, 2012.
  • C. Cattani, “Fractional calculus and Shannon wavelet,” Mathematical Problems in Engineering, vol. 2012, Article ID 502812, 26 pages, 2012.
  • M. A. E. Herzallah, A. M. A. El-Sayed, and D. Baleanu, “On the fractional-order diffusion-wave process,” Romanian Journal in Physics, vol. 55, no. 3-4, pp. 274–284, 2010.
  • M. A. E. Herzallah and D. Baleanu, “Existence of a periodic mild solution for a nonlinear fractional differential equation,” Computers and Mathematics with Applications, vol. 64, no. 10, pp. 3059–3064, 2012.
  • M. A. E. Herzallah, M. El-Shahed, and D. Baleanu, “Mild and strong solutions for a fractional nonlinear Neumann boundary value problem,” Journal of Computational Analysis and Applications, vol. 15, no. 2, pp. 341–352, 2013.
  • M. A. E. Herzallah, “Mild and strong solutions to few types of fractional order nonlinear equations with periodic boundary conditions,” Indian Journal of Pure and Applied Mathematics, vol. 43, no. 6, pp. 619–635, 2012.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, NJ, USA, 2000.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematical Studies, Elsevier, Amsterdam, The Netherlands, 2006.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applcations, Gordon and Breach, New York, NY, USA, 1993.
  • B. C. Dhage and V. Lakshmikantham, “Basic results on hybrid differential equations,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 3, pp. 414–424, 2010.
  • B. Dhage and N. Jadhav, “Basic results in the theory of hybrid differential equations with linear perturbations of second type,” Tamkang Journal of Mathematics, vol. 44, no. 2, pp. 171–186, 2013.
  • M. R. Ammi, E. El Kinani, and D. F. Torres, “Existence and uniqueness of solutions to functional integro-differential fractional equations,” Electronic Journal of Differential Equations, vol. 2012, no. 103, pp. 1–9, 2012.
  • H. Lu, S. Sun, D. Yang, and H. Teng, “Theory of fractional hybrid differential equations with linear perturbations of second type,” Boundary Value Problems, vol. 2013, article 23, 2013.
  • Y. Zhao, S. Sun, Z. Han, and Q. Li, “Theory of fractional hybrid differential equations,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1312–1324, 2011.
  • B. C. Dhage, “A fixed point theorem in Banach algebras involving three operators with applications,” Kyungpook Mathematical Journal, vol. 44, no. 1, pp. 145–155, 2004.
  • B. C. Dhage, “On a fixed point theorem in Banach algebras with applications,” Applied Mathematics Letters, vol. 18, no. 3, pp. 273–280, 2005.
  • B. C. Dhage, “Nonlinear functional boundary value problems in Banach algebras involving Caratheodories,” Kyungpook Mathematical Journal, vol. 46, no. 4, pp. 527–541, 2006. \endinput