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On the oscillation of fractional differential equations

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Abstract

In this paper we initiate the oscillation theory for fractional differential equations. Oscillation criteria are obtained for a class of nonlinear fractional differential equations of the form

$D_a^q x + f_1 (t,x) = v(t) + f_2 (t,x),\mathop {\lim }\limits_{t \to a} J_a^{1 - q} x(t) = b_1 $

, where D q a denotes the Riemann-Liouville differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo’s differential operator.

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References

  1. R.P. Agarwal, S.R. Grace, and D. O’Regan, Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations. Kluwer, Dordrecht, 2002.

    MATH  Google Scholar 

  2. M. Caputo, Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. R. Astr. Soc., 13 (1967), 529–539; Reprinted in: Fract. Calc. Appl. Anal. 11, No 1 (2008), 3–14.

    Article  Google Scholar 

  3. K. Diethelm and N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl., 265 (2002), 229–248.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Diethelm, The Analysis of Fractional Differential Equations. Springer, Berlin, 2010.

    Book  MATH  Google Scholar 

  5. K. Diethelm and A.D. Freed, On the solution of nonlinear fractional differential equations used in the modelling of viscoplasticity. In: F. Keil, W. Mackens, H. Vob and J. Werther (Eds.), Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer, Heidelberg, 1999, 217–224.

    Google Scholar 

  6. N.J. Ford, M. Luisa Morgado, Fractional boundary value problems: Analysis and numerical methods. Fract. Calc. Appl. Anal. 14, No 4 (2011), 554–567; DOI: 10.2478/s13540-011-0034-4; at http://www.springerlink.com/content/1311-0454/14/4/

    MathSciNet  Google Scholar 

  7. W.G. Glöckle and T.F. Nonnenmacher, A fractional calculus approach to self similar protein dynamics. Biophys. J. 68 (1995), 46–53.

    Article  Google Scholar 

  8. J.R. Graef, L. Kong, B. Yang, Positive solutions for a semipositone fractional boundary value problem with a forcing term. Fract. Calc. Appl. Anal. 15, No 1 (2012), 8–24; DOI: 10.2478/s13540-012-0002-7; at: http://www.springerlink.com/content/1311-0454/14/4/

    Google Scholar 

  9. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies 204, Elsevier, Amsterdam, 2006.

    Book  MATH  Google Scholar 

  10. R. Metzler, S. Schick, H.G. Kilian and T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys., 103 (1995), 7180–7186.

    Article  Google Scholar 

  11. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.

    MATH  Google Scholar 

  12. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, 1993.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. of Engineering Mathematics, Faculty of Engineering, Cairo University, Oman, Giza, 12221, Egypt

    Said R. Grace

  2. Dept. of Mathematics, Texas A & M University — Kingsville, Kingsville, TX, 78363, USA

    Ravi P. Agarwal

  3. School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

    Patricia J.Y. Wong

  4. Dept. of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Ağacık Zafer

Authors
  1. Said R. Grace
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  2. Ravi P. Agarwal
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  3. Patricia J.Y. Wong
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  4. Ağacık Zafer
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Correspondence to Said R. Grace.

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Grace, S.R., Agarwal, R.P., Wong, P.J. et al. On the oscillation of fractional differential equations. fcaa 15, 222–231 (2012). https://doi.org/10.2478/s13540-012-0016-1

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  • DOI: https://doi.org/10.2478/s13540-012-0016-1

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