International Journal of Differential Equations

Existence and Uniqueness Theorem of Fractional Mixed Volterra-Fredholm Integrodifferential Equation with Integral Boundary Conditions

Shayma Adil Murad, Hussein Jebrail Zekri, and Samir Hadid

Full-text: Open access


We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel'skii fixed-point theorem.

Article information

Int. J. Differ. Equ., Volume 2011, Special Issue (2011), Article ID 304570, 15 pages.

Received: 7 May 2011
Accepted: 24 May 2011
First available in Project Euclid: 26 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Murad, Shayma Adil; Zekri, Hussein Jebrail; Hadid, Samir. Existence and Uniqueness Theorem of Fractional Mixed Volterra-Fredholm Integrodifferential Equation with Integral Boundary Conditions. Int. J. Differ. Equ. 2011, Special Issue (2011), Article ID 304570, 15 pages. doi:10.1155/2011/304570.

Export citation


  • M. Amairi, M. Aoun, S. Najar, and M. N. Abdelkrim, “A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2162–2168, 2010.
  • Z. Drici, F. A. McRae, and J. V. Devi, “Fractional differential equations involving causal operators,” Communications in Applied Analysis., vol. 14, no. 1, pp. 81–88, 2010.
  • 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, “Existence results for fractional boundary value problem,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 33-36, pp. 1767–1774, 2008.
  • S. M. Momani, “Local and global existence theorems on fractional integro-differential equations,” Journal of Fractional Calculus, vol. 18, pp. 81–86, 2000.
  • S. M. Momani and S. B. Hadid, “On the inequalities of integro-differential fractional equations,” International Journal of Applied Mathematics, vol. 12, no. 1, pp. 29–37, 2003.
  • B. Ahmad, A. Alsaedi, and B. S. Alghamdi, “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions,” Nonlinear Analysis Real world Applications, vol. 9, no. 4, pp. 1727–1740, 2008.
  • H L. Tidke, “Existence of global solutions to nonlinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions,” Electronic Journal of Differential Equations, vol. 2009, pp. No. 55–7, 2009.
  • B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, 11 pages, 2009.
  • G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis: Theory, Method and Applications, vol. 70, no. 5, pp. 1873–1876, 2009.
  • A. Anguraj, P. Karthikeyan, and J. J. Trujillo, “Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition,” Advances in Difference Equations, vol. 2011, Article ID 690653, 12 pages, 2011.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
  • M. A. Krasnosel'skiĭ, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, no. 1(63), pp. 123–127, 1955.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.