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On the Oscillatory and Asymptotic Behavior of Solutions of Certain Integral Equations

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Abstract

We present the conditions under which every positive solution x(t) of the integral equation

$$x(t) = a(t) + \int\nolimits_c^t(t - s)^{\alpha-1}[e(s) + k(t, s)f(s, x(s))]\,{\rm d}s,\quad c > 1,\,\alpha > 0$$

Satisfies

$$x(t) = O(a(t)) \quad {\rm as}\quad t \to \infty,\quad {\rm i.e.,}\,{\mathop {\rm lim \,sup}\limits_{t \to \infty}} \frac {x(t)}{a(t)} < \infty,$$

From the obtained results, we derive a technique which can be applied to some related integral equations that are equivalent to certain fractional differential equations of Caputo derivative of any order. We also establish new oscillation criteria for such fractional differential equations.

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References

  1. Bohner M., Grace S.R., Sultana N.: Asymptotic behavior of nonoscillatory solutions of higher order integro dynamic equations. Opuscula Math. 34(1), 5–14 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Caputo M.: Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. R. Astronom. Soc. 13, 529–535 (1967)

    Article  MathSciNet  Google Scholar 

  3. Bpleanu, D., Machado, J.A.T., Luo, A.C.-J.: Fractional dynamics and control, Springer (2012)

  4. Diethelm, K.: The analysis of fractional differential equations. Springer, Berlin (2010)

  5. Furati, K.M., Tatar, N.-E.: Power-type estimates for a nonlinear fractional differential equations. Nonlinear Anal. TMA 62, 1025–1036 (2005)

  6. Grace S.R., Zafer A.: Oscillatory behavior of integro-dynamic and integral equations on time scales. Appl. Math. Lett. 28, 47–52 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Grace, S.R., Graef, J.R., Zafer, A.: Oscillation of integro-dynamic equations on time scales. Appl. Math. Lett. 26, 383–386 (2013)

  8. Grace, S.R., Greaf, J.R., Panigrahi, S.,Tunc, E.: On the oscillatory behavior of volterra integral equations on time-scales. Pan Am. Math. J. 23(2), 35–41 (2013)

  9. Grace S.R., Agarwal R.P., Wong P.J.Y., Zafer A.: On the oscillation of fractional differential equations. Fract. Calc. Appl. Anal. 15(2), 222–231 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hardy, G.H., Littlewood, I.E., Polya, G.: Inequalities. University Press, Cambridge (1959)

  11. Lakshmikantham, V., Leela, S., Vaaundhara Devi, J.: Theory of fractional dynamic systems. Cambridge Scientific Publishers (2009)

  12. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam (2006)

  13. Medved M.: Asymptotic integration of some classes of fractional differential equations. Tarta Mt. Math. 54, 119–132 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Miller, K.S., Ross, B.: An Introduction to the fractional calculus and differential equations. Wiley, New York (1993)

  15. Ma, Q.-H., Pečarić, J., Zhang, J.-M.: Integral inequalities of systems and the estimate for solutions of certain nonlinear two-dimensional fractional differential systems. Comput. Math. Appl.

  16. Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)

  17. Prudnikov, A.P., Brychkov, Zu.A., Marichev, O.I.: Integral and Series. In: Elementary Functions, vol. 1, Nauka, Moscow (1981, in Russian)

  18. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: theory and applications. Gordon and Breach Science Publishers, New York (1993)

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Authors and Affiliations

  1. Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza, 12613, Egypt

    Said R. Grace

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  1. Said R. Grace
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Correspondence to Said R. Grace.

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Dedicated to the memory of my mother

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Grace, S.R. On the Oscillatory and Asymptotic Behavior of Solutions of Certain Integral Equations. Mediterr. J. Math. 13, 2721–2732 (2016). https://doi.org/10.1007/s00009-015-0649-5

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  • DOI: https://doi.org/10.1007/s00009-015-0649-5

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