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On the oscillatory behavior of solutions of higher order nonlinear fractional differential equations

  • Said R. Grace and Ercan Tunç EMAIL logo
Published/Copyright: June 29, 2017
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 25 Issue 3

Abstract

The study of oscillation theory for fractional differential equations has been initiated by Grace et al. [5]. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form DarC⁢x⁢(t)=e⁢(t)+f⁢(t,x⁢(t)),t>0,a>1, where r=α+n-1,α∈(0,1), and n≥1 is a natural number. We also present the conditions under which all solutions of this equation are asymptotic to tn-1 as t→∞.

Keywords: Asymptotic behavior; oscillation; nonoscillatory solution; Caputo derivative; higher order, fractional differential equations
MSC 2010: 34E10; 34A34

References

[1] D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer, New York, 2012. 10.1007/978-1-4614-0457-6Search in Google Scholar

[2] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. Royal Astronom. Soc. 13 (1967), no. 5, 529–535. 10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[3] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Math. 2004, Springer, Berlin, 2010. 10.1007/978-3-642-14574-2Search in Google Scholar

[4] K. M. Furati and N. Tatar, Power-type estimates for a nonlinear fractional differential equation, Nonlinear Anal. 62 (2005), no. 6, 1025–1036. 10.1016/j.na.2005.04.010Search in Google Scholar

[5] S. R. Grace, R. P. Agarwal, P. J. Y. Wong and A. Zafer, On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal. 15 (2012), no. 2, 222–231. 10.2478/s13540-012-0016-1Search in Google Scholar

[6] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1952. Search in Google Scholar

[7] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science B.V., Amsterdam, 2006. Search in Google Scholar

[8] V. Lakshmikantham, S. Leela and J. Vaaundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, 2009. Search in Google Scholar

[9] Q.-H. Ma, J. Pečarić and J.-M. Zhang, Integral inequalities of systems and the estimate for solutions of certain nonlinear two-dimensional fractional differential systems, Comput. Math. Appl. 61 (2011), no. 11, 3258–3267. 10.1016/j.camwa.2011.04.008Search in Google Scholar

[10] M. Medved’, Asymptotic integration of some classes of fractional differential equations, Tatra Mt. Math. Publ. 54 (2013), 119–132. 10.2478/tmmp-2013-00010Search in Google Scholar

[11] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. Search in Google Scholar

[12] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, Inc., San Diego, 1999. Search in Google Scholar

[13] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series (in Russian), Nauka, Moscow, 1981. Search in Google Scholar

[14] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993. Search in Google Scholar

[15] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), no. 2, 1075–1081. 10.1016/j.jmaa.2006.05.061Search in Google Scholar

Received: 2015-06-06
Accepted: 2015-09-29
Published Online: 2017-06-29
Published in Print: 2018-09-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Semilinear fractional order integro-differential inclusions with infinite delay
  3. Quasi-nilpotent perturbations of the generalized Kato spectrum
  4. Construction and numerical resolution of high-order accuracy decomposition scheme for a quasi-linear evolution equation
  5. Relations between BV*(q;α) and Λ*BVp classes of functions
  6. On the constancy of signs and order of magnitude of Fourier–Haar coefficients
  7. On the oscillatory behavior of solutions of higher order nonlinear fractional differential equations
  8. Optimal control problem for the equation of vibrations of an elastic plate
  9. Generalized Hausdorff capacities and their applications
  10. Approximation by bivariate (p,q)-Baskakov–Kantorovich operators
  11. A hydromagnetic flow through porous medium near an accelerating plate in the presence of magnetic field
  12. A note on the Borel types of some small sets
  13. On some boundary value problems for the heat equation in a non-regular type of a prism of ℝN+1
  14. Some weighted integral inequalities for differentiable h-preinvex functions
  15. Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras
  16. On Baer invariants of pairs of groups
  17. The Arzelà–Ascoli theorem by means of ideal convergence
  18. Absolute convergence of multiple Fourier series of a function of p(n)-Λ-BV
https://doi.org/10.1515/gmj-2017-0026
Keywords for this article
Asymptotic behavior; oscillation; nonoscillatory solution; Caputo derivative; higher order, fractional differential equations

Articles in the same Issue

  1. Frontmatter
  2. Semilinear fractional order integro-differential inclusions with infinite delay
  3. Quasi-nilpotent perturbations of the generalized Kato spectrum
  4. Construction and numerical resolution of high-order accuracy decomposition scheme for a quasi-linear evolution equation
  5. Relations between BV*(q;α) and Λ*BVp classes of functions
  6. On the constancy of signs and order of magnitude of Fourier–Haar coefficients
  7. On the oscillatory behavior of solutions of higher order nonlinear fractional differential equations
  8. Optimal control problem for the equation of vibrations of an elastic plate
  9. Generalized Hausdorff capacities and their applications
  10. Approximation by bivariate (p,q)-Baskakov–Kantorovich operators
  11. A hydromagnetic flow through porous medium near an accelerating plate in the presence of magnetic field
  12. A note on the Borel types of some small sets
  13. On some boundary value problems for the heat equation in a non-regular type of a prism of ℝN+1
  14. Some weighted integral inequalities for differentiable h-preinvex functions
  15. Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras
  16. On Baer invariants of pairs of groups
  17. The Arzelà–Ascoli theorem by means of ideal convergence
  18. Absolute convergence of multiple Fourier series of a function of p(n)-Λ-BV
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