Abstract and Applied Analysis

On Solutions of a Nonlinear Erdélyi-Kober Integral Equation

Nurgali K. Ashirbayev, Józef Banaś, and Raina Bekmoldayeva

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


We conduct some investigations concerning the solvability of a nonlinear integral equation of Erdélyi-Kober type. To facilitate our study we will first consider a nonlinear integral equation of Volterra-Stieltjes type. Since the mentioned Erdélyi-Kober integral equation turns out to be a special case of that of Volterra-Stieltjes type, we can apply the obtained results to the Erdélyi-Kober integral equation. Examples illustrating the obtained results will be also included.

Article information

Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 184626, 7 pages.

First available in Project Euclid: 6 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Ashirbayev, Nurgali K.; Banaś, Józef; Bekmoldayeva, Raina. On Solutions of a Nonlinear Erdélyi-Kober Integral Equation. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 184626, 7 pages. doi:10.1155/2014/184626. https://projecteuclid.org/euclid.aaa/1412607126

Export citation


  • 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, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
  • V. Lakshmikantham, S. Leela, and J. Vasundara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, UK, 2009.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Amsterdam, The Netherlands, 1993.
  • H. M. Srivastava and R. K. Saxena, “Operators of fractional integration and their applications,” Applied Mathematics and Computation, vol. 118, no. 1, pp. 1–52, 2001.
  • J. A. Alamo and J. Rodríguez, “Operational calculus for modified Erdélyi-Kober operators,” Serdica Bulgaricae Mathematicae Publicationes, vol. 20, no. 3-4, pp. 351–363, 1994.
  • H. H. Hashem and M. S. Zaki, “Carathéodory theorem for quadratic integral equations of Erdélyi-Kober type,” Journal of Fractional Calculus and Applications, vol. 4, no. 1, pp. 56–72, 2013.
  • M. A. Darwish and K. Sadarangani, “On Erdélyi-Kober type quadratic integral equation with linear modification of the argument,” Applied Mathematisc and Computation, vol. 238, pp. 30–42, 2014.
  • J. R. Wang, C. Zhu, and M. Feckan, “Solvability of fully nonlinear functional equations involving Erdélyi-Kober fractional integrals on the unbounded interval,” Optimization, 2014.
  • J. Appell, J. Banaś, and N. Merentes, Bounded Variation and Around, vol. 17 of De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 2014.
  • I. P. Natanson, Theory of Functions of a Real Variable, Ungar, New York, NY, USA, 1960.
  • N. Danford and J. T. Schwartz, Linear Operators, International Publishing, Leyden, The Netherlands, 1963.
  • A. Erdélyi, “On fractional integration and its application to the theory of Hankel transforms,” The Quarterly Journal of Mathematics, vol. 11, p. 293–-303, 1940.
  • A. Erdélyi and H. Kober, “Some remarks on Hankel transforms,” The Quarterly Journal of Mathematics, vol. 11, pp. 212–221, 1940.
  • H. Kober, “On fractional integrals and derivatives,” The Quarterly Journal of Mathematics, vol. 11, pp. 193–211, 1940.
  • J. Banaś and T. Zaj\kac, “A new approach to the theory of functional integral equations of fractional order,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 375–387, 2011.
  • T. Zaj\kac, “Solvability of fractional integral equations on an unbounded interval through the theory of Volterra-Stieltjes integral equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 33, no. 1, pp. 65–85, 2014. \endinput