Sufficient conditions on existence of solution for nonlinear fractional iterative integral equation

Volume 10, Issue 2, pp 368--376 http://dx.doi.org/10.22436/jnsa.010.02.03

Publication Date: February 20, 2017 Submission Date: September 28, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2541 Downloads
• 5829 Views

Authors

Faten H. M. Damag - Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia. Adem Kilicman - Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia.

Abstract

In this article, we study nonlinear quadratic iterative integral equations and establish sufficient conditions for the existence of Volterra solutions for fractional iterative integral equations and solvency in Banach space and \(C_{\ell,\beta}\). In the present work we use the principle of contraction, Schaefer’s fixed point theorem and the non-expansive operator method as essential tools. In this study we consider Riemann-Liouville differential operator and prove some related theorems, further provide an example as an application.

Share and Cite

Keywords

• Fractional integral equation
• existence of solution
• Schaefer’s fixed point theorem
• non-expansive operator.

MSC

•  39A12
•  37N25
•  35E15

References

• [1] A. Atangana, R. T. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Adv. Difference Equ., 2016 (2016), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447–454.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] V. Berinde, Iterative approximation of fixed points, Second edition, Lecture Notes in Mathematics, Springer, Berlin (2007)
  • View Article
  • Google Scholar


• [4] V. Berinde, Existence and approximation of solutions of some first order iterative differential equations, Miskolc Math. Notes, 11 (2010), 13–26.
  • MathSciNet
  • Google Scholar


• [5] S. S. Cheng, J.-G. Si, X.-P.Wang, An existence theorem for iterative functional differential equations, Acta Math. Hungar., 94 (2002), 1–17.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] F. H. Damag, A. Kılıçman, R. A. A. Abdulghafor, Approximate solutions and existence result for some integral equation with modified argument of fractional order, Adv. Difference Equ., (2016), submitted


• [7] F. H. Damag, A. Kılıçman, R. W. Ibrahim, Approximate solutions for non-linear iterative fractional differential equations, Innovations Through Mathematical and Statistical Research, Proceedings of the 2nd International Conference on Mathematical Sciences and Statistics (ICMSS2016) , AIP Publishing,1739 ((2016). )
  • View Article
  • Google Scholar


• [8] F. H. Damag, A. Kılıçman, R. W. Ibrahim, Findings of fractional iterative differential equations involving first order derivative, Int. J. Appl. Comput. Math., (2016), 1–10.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] A. El-Sayed, H. Hashem, Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra, Fract. Calc. Appl. Anal., 16 (2013), 816–826.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] A. Granas, J. Dugundji, Fixed point theory, Springer Science and Business Media, (2013)
  • Google Scholar


• [11] R.W. Ibrahim, A. Kılıçman, F. H. Damag, Existence and uniqueness for a class of iterative fractional differential equations, Adv. Difference Equ., 2015 (2015 ), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [12] R. W. Ibrahim, A. Kılıçman, F. H. Damag, Extremal solutions by monotone iterative technique for hybrid fractional differential equations, Turkish J. Anal. Number Theory, 4 (2016), 60–66.


• [13] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65–71.
  • View Article
  • Google Scholar


• [14] M. Lauran , Existence results for some integral equation with modified argument, Gen. Math., 19 (2011), 85–92.
  • MathSciNet
  • Google Scholar
  • MATH


• [15] M. Lauran, Existence results for some nonlinear integral equations, Miskolc Math. Notes, 13 (2012), 67–74.
  • MathSciNet
  • Google Scholar


• [16] M. Lauran, Solution of first iterative differential equations, An. Univ. Craiova Ser. Mat. Inform, 40 (2013), 45–51.
  • MathSciNet
  • Google Scholar


• [17] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993)
  • MathSciNet
  • Google Scholar


• [18] D. O’Regan, Existence results for nonlinear integral equations, J. Math. Anal. Appl., 192 (1995), 705–726.
  • View Article
  • Google Scholar


• [19] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)


• [20] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolskiĭ, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
  • Google Scholar


• [21] J.-R. Wang, M. Fečkan, Y. Zhou, Fractional order iterative functional differential equations with parameter, Appl. Math. Model., 37 (2013), 6055–6067.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] P.-P. Zhang, X.-B. Gong, Existence of solutions for iterative differential equations, Electron, J. Differential Equations, 2014 (2014 ), 10 pages.
  • MathSciNet
  • Google Scholar