Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero

Covered by:
Web of Science - Emerging Sources Citation Index

CiteScore 2018: 0.47
SCImago Journal Rank (SJR) 2018: 0.265
Source Normalized Impact per Paper (SNIP) 2018: 0.714

Mathematical Citation Quotient (MCQ) 2018: 0.17

ICV 2018: 121.16

Open Access
See all formats and pricing
More options …
Volume 51, Issue 1


Global attractivity for Volterra type Hadamard fractional integral equations in Fréchet spaces

Saïd Abbas
  • Laboratory of Mathematics, Geometry, Analysis, Control and Applications, Tahar Moulay University of Saïda, P.O. Box 138, EN-Nasr, 20000 Saïda, Algeria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ravi P. Agarwal / Mouffak Benchohra
  • Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbes 22000, Algeria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Farida Berhoun
  • Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbes 22000, Algeria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-05-31 | DOI: https://doi.org/10.1515/dema-2018-0009


In this paper, we present some results concerning the existence and the attractivity of solutions for some functional integral equations of Hadamard fractional order. We use an extension of the Burton-Kirk fixed point theorem in Fréchet spaces.

Keywords: functional integral equation; Hadamard integral of fractional order; solution; attractivity; Fréchet space; fixed point

MSC 2010: Primary 26A33; Secondary 45G05; 45M10


  • [1] Abbas S., Benchohra M., N’Guérékata G. M., Topics in Fractional Differential Equations, Developments in Mathematics, 27, Springer, New York, 2012Google Scholar

  • [2] Abbas S., Benchohra M., N’Guérékata G. M., Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015Google Scholar

  • [3] Baleanu D., Diethelm K., Scalas E., Trujillo J. J., Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012Google Scholar

  • [4] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006Google Scholar

  • [5] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993Google Scholar

  • [6] Lakshmikantham V., Leela S., Vasundhara J., Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009Google Scholar

  • [7] Samko S. G., Kilbas A. A.,Marichev O. L., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993Google Scholar

  • [8] Butzer P. L., Kilbas A. A., Trujillo J. J., Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J.Math. Anal. Appl., 2002, 269, 1-27Google Scholar

  • [9] Butzer P. L., Kilbas A. A., Trujillo J. J., Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl., 2002, 270, 1-15Google Scholar

  • [10] Pooseh S., Almeida R., Torres D., Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim., 2012, 33(3), 301-319Web of ScienceGoogle Scholar

  • [11] Abbas A., Alaidarous E., Benchohra M., Nieto J. J, Existence and stability of solutions for Hadamard-Stieltjes fractional integral equations, Discrete Dyn. Nat. Soc., 2015, Art. ID 317094Google Scholar

  • [12] Adjabi Y., Jarad F., Baleanu D., Abdeljawad T., On Cauchy problems with Caputo Hadamard fractional derivatives, J. Comput. Anal. Appl., 2016, 21(4), 661-681Google Scholar

  • [13] Aljoudi S., Ahmad B., Nieto J. J., Alsaedi A., A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions, Chaos Solitons Fractals, 2016, 91, 39-46Web of ScienceCrossrefGoogle Scholar

  • [14] Benchohra M., Bouriah S., Nieto J. J., Existence of periodic solutions for nonlinear implicit Hadamard’s fractional differential equations, Rev. R. Acad. Cienc. Exactas, Fís. Nat. Ser. A Math. RACSAM, 2018, 112, 25-35Google Scholar

  • [15] Gambo Y. Y., Jarad F., Baleanu D., Abdeljawad T., On Caputo modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2014, 2014:10Google Scholar

  • [16] Wang G., Pei K., Baleanu D., Explicit iteration to Hadamard fractional integro-differential equations on infinite domain, Adv. Difference Equ., 2016, 2016:299Google Scholar

  • [17] Abbas S., Benchohra M., Nonlinear quadratic Volterra Riemann-Liouville integral equations of fractional order, Nonlinear Anal. Forum, 2012, 17, 1-9Google Scholar

  • [18] Abbas S., Benchohra M., Fractional order Riemann-Liouville integral equations with multiple time delay, Appl. Math. ENotes, 2012, 12, 79-87Google Scholar

  • [19] Abbas S., Benchohra M., Henderson J., On global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order, Comm. Appl. Nonlinear Anal., 2012, 19, 79-89Google Scholar

  • [20] Abbas S., Benchohra M., Vityuk A. N.,On fractional order derivatives and Darboux problem for implicit differential equations, Fract. Calc. Appl. Anal., 2012, 15(2), 168-182Web of ScienceGoogle Scholar

  • [21] Banaś J., Dhage B. C., Global asymptotic stability of solutions of a functional integral equation, Nonlinear Anal., 2008, 69(7), 1945-1952Google Scholar

  • [22] Banaś J., Rzepka B., On existence and asymptotic stability of solutions of a nonlinear integral equation, J.Math. Anal. Appl., 2003, 284, 165-173Google Scholar

  • [23] Banaś J., Zając T., Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity, Nonlinear Anal., 2009, 71, 5491-5500Google Scholar

  • [24] Banaś J., Zając T., A new approach to the theory of functional integral equations of fractional order, J. Math. Anal. Appl., 2011, 375, 375-387Web of ScienceCrossrefGoogle Scholar

  • [25] Darwish M. A., Henderson J., O’Regan D., Existence and asymptotic stability of solutions of a perturbed fractional functional integral equations with linear modification of the argument, Bull. Korean Math. Soc., 2011, 48(3), 539-553CrossrefGoogle Scholar

  • [26] Pachpatte B. G., On Volterra-Fredholm integral equation in two variables, Demonstratio Math., 2007, XL(4), 839-852Google Scholar

  • [27] Pachpatte B. G., On Fredholm type integral equation in two variables, Differ. Equ. Appl., 2009, 1, 27-39Google Scholar

  • [28] Hadamard J., Essai sur L’étude des Fonctions Données par Leur Développment de Taylor, J. Pure Appl. Math., 1892, 4(8), 101-186Google Scholar

  • [29] Frigon M., Granas A., Théorèmes d’Existence pour des Inclusions Différentielles sans Convexité, C. R. Acad. Sci. Paris, Ser. I, 1990, 310, 819-822Google Scholar

  • [30] Avramescu C., Some remarks on a fixed point theorem of Krasnoselskii, Electron. J. Qual. Theory Differ. Equ., 2003, 5, 1-15Google Scholar

About the article

Received: 2017-01-10

Accepted: 2018-04-09

Published Online: 2018-05-31

Citation Information: Demonstratio Mathematica, Volume 51, Issue 1, Pages 131–140, ISSN (Online) 2391-4661, DOI: https://doi.org/10.1515/dema-2018-0009.

Export Citation

© 2018 Saïd Abbas et al. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in