On the Asymptotic Behavior of Nonoscillatory Solutions of Certain Fractional Differential Equations

  • Said R. Grace


We present the conditions under which every nonoscillator solution x(t) of the forced fractional differential equation
$$\begin{aligned} ^{\mathrm{C}}D_{\mathrm{c}}^{\alpha } y ( t ) = e ( t ) +f ( {t, x ( t )} ), c > 1,\alpha \in ( {0,1} ), \quad \mathrm{{and}} \,\, \delta \ge 1, \end{aligned}$$
where \(y(t)= ( {a(t) ( {{x}'(t)} )^{\delta }})^{\prime },c_0 =\frac{y(c)}{\Gamma (1)}= y(c)\), is a real constant which satisfies
$$\begin{aligned} |x(t)|=O\left( {t^{1/\delta }e^{t}\int _{\mathrm{c}}^{t} {a^{-1/\delta }} (s)\mathrm{d}s} \right) , \quad t \rightarrow \infty \end{aligned}$$
It is shown that the technique can be applied to some related fractional differential equations. Examples are inserted to illustrate the relevance of the obtained results.


Integro-differential equation fractional differential equations asymptotic behavior nonoscillatory solutions 

Mathematics Subject Classification

34E10 34A34 


  1. 1.
    Băleanu, D., Machado, J.A.T., Luo, A.C.-J.: Fractional Dynamics and Control. Springer, Berlin (2012)CrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brestovanska, E., Medved, M.: Asymptotic behavior of solutions to second-order differential equations with fractional derivative perturbations. Electron. J. Qual. Theory Differ. Equ. 2014(201), 1.10 (2014)Google Scholar
  4. 4.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. R. Astron. Soc. 13, 529–535 (1967)CrossRefGoogle Scholar
  5. 5.
    Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Furati, K.M., Tatar, N.-E.: Power-type estimates for a nonlinear fractional differential equations. Nonlinear Anal. TMA 62, 1025–1036 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grace, S.R., Zafer, A.: Oscillatory behavior of integro-dynamic and integral equations on time scales. Appl. Math. Lett. 28, 47–52 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grace, S.R., Graef, J.R., Zafer, A.: Oscillation of integro-dynamic equations on time scales. Appl. Math. Lett. 26, 383–386 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grace, S.R.: On the asymptotic behavior of positive solutions of certain fractional differential equations. In: Mathematical Problems in Engineering Volume 2015, Article ID 945347, pp 7Google Scholar
  12. 12.
    Hardy, G.H., Littlewood, I.E., Polya, G.: Inequalities. University Press, Cambridge (1959)zbMATHGoogle Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    Lakshmikantham, V., Leela, S., Vaaundhara Devi, J.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)zbMATHGoogle Scholar
  15. 15.
    Ma, Q.-H., Pecari-c, J., Zhang, J.-M.: Integral inequalities of systems and the estimate for solutions of certain nonlinear two-dimensional fractional differential systems. Comput. Math. Appl. 61, 3258–3267 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  17. 17.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  18. 18.
    Prudnikov, A.P., Brychkov, Zu.A., Marichev, O.I.: Integral and series. In: Elementary Functions, vol. 1. Nauka, Moscow, 1981 (in Russian) Google Scholar
  19. 19.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, New York (1993)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering Mathematics, Faculty of EngineeringCairo UniversityGizaEgypt

Personalised recommendations