Skip to main content
SpringerLink
Log in

Fuzzy fractional integral equations under compactness type condition

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

In this paper we study a fuzzy fractional integral equation. The fractional derivative is considered in the sense of Riemann-Liouville and we establish existence of the solutions of fuzzy fractional integral equations using the Hausdorff measure of noncompactness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 74 (2010), 2859–2862.

    Article  MathSciNet  Google Scholar 

  2. R.P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Difference Equ. (2009) Article ID 981728, 47pp.

  3. R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundry value problems for nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109 (2010), 973–1033.

    Article  MathSciNet  MATH  Google Scholar 

  4. R.P. Agarwal, Y. Zhou, J. Wang, X. Luo, Fractional functional differential equations with causal operators in Banach spaces. Mathematical and Compututer Modelling 54 (2011), 1440–1452.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Alikhani, F. Bahrami, A. Jabbari, Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations. Nonlinear Anal. 75 (2012), 1810–1821.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Allahviranloo, S. Salahshour, S. Abbasbandy, Explicit solutions of fractional differential equations with uncertainty. Soft Computing 16, No 2 (2012), 297–302.

    Article  Google Scholar 

  7. S. Arshad, V. Lupulescu, On the fractional differential equations with uncertainty. Nonlinear Anal. 74 (2011), 3685–3693.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Arshad, V. Lupulescu, Fractional differential equation with fuzzy initial condition. Electronic J. of Differential Equations 2011 (2011), 1–8.

    MathSciNet  Google Scholar 

  9. M. Benchohra, M. A. Darwish, Existence and uniqueness theorem for fuzzy integral equations of fractional order. Communications in Applied Analysis 12 (2008), 13–22.

    MathSciNet  MATH  Google Scholar 

  10. W. Congxin, S. Shiji, Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions. Inform. Sci. 108 (1998), 123–134.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, Berlin-Heidelberg, 1985.

    Book  MATH  Google Scholar 

  12. P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets. World Scientific, Singapore, 1994.

    MATH  Google Scholar 

  13. K. Diethelm, An efficient parallel algorithm for the numerical solution of fractional differential equations. Fract. Calc. and Appl. Anal. 14, No 3 (2011), 475–490; DOI: 10.2478/s13540-011-0029-1; at http://www.springerlink.com/content/1311-0454/14/3/.

    MathSciNet  Google Scholar 

  14. K. Diethelm, The Analysis of Fractional Differential Equations. Springer, 2004.

  15. K. Diethelm, The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus. Fract. Calc. and Appl. Anal. 15, No 2 (2012), 304–313; DOI: 10.2478/s13540-012-0022-3; at http://www.springerlink.com/content/1311-0454/15/2/.

    MathSciNet  Google Scholar 

  16. T. Donchev, A. Nosheen, On the solution set of fuzzy systems. In: Nonlinear Anal., 2012.

  17. A.M.A. El-Sayed, A.-G. Ibrahim, Set-valued integral equations of fractional-orders. Applied Mathematics and Computation 118 (2001), 113–121.

    Article  MathSciNet  MATH  Google Scholar 

  18. S.R. Grace, R.P. Agarwal, P.J.Y. Wong and A. Zafer, On the oscillation of fractional differential equations. Fract. Calc. and Appl. Anal. 15, No 2 (2012), 222–231; DOI: 10.2478/s13540-012-0016-1; at http://www.springerlink.com/content/1311-0454/15/2/.

    MathSciNet  Google Scholar 

  19. S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, 1997.

  20. O. Kaleva, The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems 35 (1990), 389–396.

    Article  MathSciNet  MATH  Google Scholar 

  21. D. Kandilakis, N.S. Papageorgiou, On the properties of the Aumann integral with applications to differential inclusions and control systems. Czech. Math. Journ. 39 (1989), 1–15.

    MathSciNet  Google Scholar 

  22. 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, New York, 2006.

  23. M. Kisielewicz, Multivalued differential equations in separable Banach spaces. J. Opt. Theory Appl. 37 (1982), 231–249.

    Article  MathSciNet  MATH  Google Scholar 

  24. V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations. Nonlinear Anal. 69 (2008), 2677–2682.

    Article  MathSciNet  MATH  Google Scholar 

  25. V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions. Taylor & Francis, London, 2003.

    Book  MATH  Google Scholar 

  26. V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces. Pergamon Press, New York, 1969.

    Google Scholar 

  27. V. Lupulescu, Causal functional differential equations in Banach spaces. Nonlinear Anal. 69 (2008), 4787–4795.

    Article  MathSciNet  MATH  Google Scholar 

  28. M.T. Malinowski, Random fuzzy differential equations under generalized Lipschitz condition. Nonlinear Analysis: Real World Applications 13, No 2 (2012), 860–881.

    Article  MathSciNet  MATH  Google Scholar 

  29. K.S. Miller, B. Ross, An introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.

    MATH  Google Scholar 

  30. C.V. Negoita, D. Ralescu, Applications of Fuzzy Sets to Systems Analysis. Wiley, New York, 1975.

  31. K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to an arbitrary order. Academic Press, New York — London, 1974.

    Google Scholar 

  32. N.S. Papageorgiou, Existence of solutions for integrodifferential inclusions in Banach spaces. Commentationes Mathematicae Universitatis Carolinae 32, No 4 (1991), 687–696.

    MathSciNet  MATH  Google Scholar 

  33. J.Y. Park, H. K. Han, Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets and Systems 105 (1999), 481–488.

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Puri, D. Ralescu, Fuzzy random variables. J. Math. Anal. Appl. 114 (1986), 409–422.

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), 1372–1381.

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Song, Q. Liu, Q. Xu, Existence and comparison theorems to Volterra fuzzy integral equations in (E n,D). Fuzzy Sets and Systems 104 (1999), 315–321.

    Article  MathSciNet  MATH  Google Scholar 

  37. H. Wang, Y. Liu, Existence results for fuzzy integral equations of fractional order. Int. Journal of Math. Analysis 5 (2011), 811–818.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University-Kingvsille, 700 University Blvd., Kingsville, TX, 78363-8202, USA

    Ravi P. Agarwal

  2. Abdus Salam School of Mathematical Sciences GC University, 68-B New Muslim Town, Lahore, Pakistan

    Sadia Arshad & Vasile Lupulescu

  3. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, University Road, Galway, Ireland

    Donal O’Regan

  4. Constantin Brancusi University, Republicii 1, 210152, Targu-Jiu, Romania

    Vasile Lupulescu

Authors
  1. Ravi P. Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sadia Arshad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vasile Lupulescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ravi P. Agarwal.

About this article

Cite this article

Agarwal, R.P., Arshad, S., O’Regan, D. et al. Fuzzy fractional integral equations under compactness type condition. fcaa 15, 572–590 (2012). https://doi.org/10.2478/s13540-012-0040-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s13540-012-0040-1

MSC 2010

Key Words and Phrases

Search

Navigation