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Oscillation criteria for a certain class of fractional order integro-differential equations

Year 2017, Volume: 46 Issue: 2, 199 - 207, 01.04.2017

Abstract

In this paper, we shall give some new results about the oscillatory behavior of nonlinear fractional order integro-differential equations with forcing term $v(t)$ of form \[ D_a^\alpha x(t)=v(t)-\int\limits_a^t K(t,s) F(s,x(s))ds, \,\, 0<\alpha <1,\,\, \lim\limits_{t\to a^+} J_a^{1-\alpha} x(t)=b_1, \]

where $v$, $K$ and $F$ are continuous functions, $b_1\in\mathbb{R}$, and $D_a^\alpha$ and $J_a^{1-\alpha}$ denote the Riemann-Liouville fractional order differential and integral operators respectively.

References

  • Alzabut, J.O., Abdeljawad, T. Sufficient conditions for the oscillation of nonlinear frac- tional difference equations, J. Fract. Calc. Appl. 5, 177-187, 2014.
  • Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J. Fractional calculus models and numerical methods. Series on Complexity, Nonlinearity and Chaos (World Scientic, 2012).
  • Baleanu, D., Mustafa, O.G., O'Regan, D. On a fractional dierential equation with infinitely many solutions, Adv. Difference. Eq. 2012:145, 6pp, 2012.
  • Baleanu, D., Mustafa, O.G., O'Regan, D. Kamenev-type result for a linear ($1+\alpha$)-order fractional differential equation, App. Math. Comput. 259, 374-378, 2015.
  • Caputo, M. Linear models of dissipation whose Q is almost frequency independent, II. Reprinted from Geophys. J. R. Astr. Soc. 13, 529-539, 1967. Fract. Calc. Appl. Anal. 11, 4-14, 2008.
  • Chen, D. X., Qu, P. X., Lan, Y.H. Forced oscillation of certain fractional differential equa- tions, Adv. Difference Equ. 125, 10pp, 2013.
  • Chen, D. X. Oscillation criteria of fractional differential equations, Adv. Difference Equ. 33, 1-18, 2012.
  • Chen, D. Oscillatory behavior of a class of fractional differential equations with damping, Universitatea Politehnica din Bucuresti Scientic Bulletin A. 75, 107-118, 2013.
  • Diethelm, K. and Freed, A.D. 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.), Scientic Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer, Heidelberg. 217-224, 1999.
  • Diethelm, K. The Analysis of Fractional Differential Equations (Springer, 2010).
  • Edwards, J.T., Ford, N.J., Simpson, A.C. The numerical solution of linear multi-term frac- tional differential equations: systems of equations, J. Comput. Appl. Math. 148, 401-418, 2002.
  • Feng, Q. and Meng, F. Oscillation of solutions to nonlinear forced fractional differential equation, Electron. J. Dierential Equations 169, 10pp, 2013.
  • Galeone, L. and Garrappa, R. Explicit methods for fractional differential equations and their stability properties, J. Comput. Appl. Math. 228, 548-560, 2009.
  • Glöckle, W.G. and Nonnenmacher, T.F. A fractional calculus approach to self similar pro- tein dynamics, Biophys. J. 68, 46-53, 1995.
  • Grace, S.R., Agarwal, R.P., Wong, P.J.Y., Zafer, A. On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal. 15, 222-231, 2012.
  • Han, Z., Zhao, Y., Sun, Y., Zhang, C. Oscillation for a class of fractional differential equation, Discrete Dyn. Nat. Soc. 2013, Article ID 390282, 6 pp, 2013.
  • Hardy, G. H., Littlewood, J. E. and Polya, G. Inequalities (Cambridge University Press, 1988).
  • Mainardi, F. Fractional calculus: some basic problems in continuum and statistical mechan- ics. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics 378 (Springer-Verlag Wien, 1997), 291-348.
  • Miller K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differ- ential Equations (JohnWiley & Sons, 1993).
  • Muslim, M. Existence and approximation of solutions to fractional differential equations, Math. Comput. Modelling. 49, 1164-1172, 2009 .
  • Öğrekçi, S. Interval oscillation criteria for functional differential equations of factional order, Adv. Difference. Eq. 2015:3, 8pp, 2015.
  • Podlubny, I. Fractional Differential Equations (Academic Press, 1999).
  • Samko, S.G., Kilbas, A.A. and Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, 1993).
  • Trigeassou, J.C., Maamri, N., Sabatier, J., Oustaloup, A. A Lyapunov approach to the stability of fractional differential equations, Signal Process. 91, 437-445, 2011.
  • Zhou, Y., Jiao, F. and Li, J. Existence and uniqueness for p-type fractional neutral differ- ential equations, Nonlinear Anal. 71, 2724-2733, 2009.
Year 2017, Volume: 46 Issue: 2, 199 - 207, 01.04.2017

Abstract

References

  • Alzabut, J.O., Abdeljawad, T. Sufficient conditions for the oscillation of nonlinear frac- tional difference equations, J. Fract. Calc. Appl. 5, 177-187, 2014.
  • Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J. Fractional calculus models and numerical methods. Series on Complexity, Nonlinearity and Chaos (World Scientic, 2012).
  • Baleanu, D., Mustafa, O.G., O'Regan, D. On a fractional dierential equation with infinitely many solutions, Adv. Difference. Eq. 2012:145, 6pp, 2012.
  • Baleanu, D., Mustafa, O.G., O'Regan, D. Kamenev-type result for a linear ($1+\alpha$)-order fractional differential equation, App. Math. Comput. 259, 374-378, 2015.
  • Caputo, M. Linear models of dissipation whose Q is almost frequency independent, II. Reprinted from Geophys. J. R. Astr. Soc. 13, 529-539, 1967. Fract. Calc. Appl. Anal. 11, 4-14, 2008.
  • Chen, D. X., Qu, P. X., Lan, Y.H. Forced oscillation of certain fractional differential equa- tions, Adv. Difference Equ. 125, 10pp, 2013.
  • Chen, D. X. Oscillation criteria of fractional differential equations, Adv. Difference Equ. 33, 1-18, 2012.
  • Chen, D. Oscillatory behavior of a class of fractional differential equations with damping, Universitatea Politehnica din Bucuresti Scientic Bulletin A. 75, 107-118, 2013.
  • Diethelm, K. and Freed, A.D. 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.), Scientic Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer, Heidelberg. 217-224, 1999.
  • Diethelm, K. The Analysis of Fractional Differential Equations (Springer, 2010).
  • Edwards, J.T., Ford, N.J., Simpson, A.C. The numerical solution of linear multi-term frac- tional differential equations: systems of equations, J. Comput. Appl. Math. 148, 401-418, 2002.
  • Feng, Q. and Meng, F. Oscillation of solutions to nonlinear forced fractional differential equation, Electron. J. Dierential Equations 169, 10pp, 2013.
  • Galeone, L. and Garrappa, R. Explicit methods for fractional differential equations and their stability properties, J. Comput. Appl. Math. 228, 548-560, 2009.
  • Glöckle, W.G. and Nonnenmacher, T.F. A fractional calculus approach to self similar pro- tein dynamics, Biophys. J. 68, 46-53, 1995.
  • Grace, S.R., Agarwal, R.P., Wong, P.J.Y., Zafer, A. On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal. 15, 222-231, 2012.
  • Han, Z., Zhao, Y., Sun, Y., Zhang, C. Oscillation for a class of fractional differential equation, Discrete Dyn. Nat. Soc. 2013, Article ID 390282, 6 pp, 2013.
  • Hardy, G. H., Littlewood, J. E. and Polya, G. Inequalities (Cambridge University Press, 1988).
  • Mainardi, F. Fractional calculus: some basic problems in continuum and statistical mechan- ics. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics 378 (Springer-Verlag Wien, 1997), 291-348.
  • Miller K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differ- ential Equations (JohnWiley & Sons, 1993).
  • Muslim, M. Existence and approximation of solutions to fractional differential equations, Math. Comput. Modelling. 49, 1164-1172, 2009 .
  • Öğrekçi, S. Interval oscillation criteria for functional differential equations of factional order, Adv. Difference. Eq. 2015:3, 8pp, 2015.
  • Podlubny, I. Fractional Differential Equations (Academic Press, 1999).
  • Samko, S.G., Kilbas, A.A. and Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, 1993).
  • Trigeassou, J.C., Maamri, N., Sabatier, J., Oustaloup, A. A Lyapunov approach to the stability of fractional differential equations, Signal Process. 91, 437-445, 2011.
  • Zhou, Y., Jiao, F. and Li, J. Existence and uniqueness for p-type fractional neutral differ- ential equations, Nonlinear Anal. 71, 2724-2733, 2009.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Serkan Aslıyüce This is me

A. Feza Güvenilir

Ağacık Zafer This is me

Publication Date April 1, 2017
Published in Issue Year 2017 Volume: 46 Issue: 2

Cite

APA Aslıyüce, S., Güvenilir, A. F., & Zafer, A. (2017). Oscillation criteria for a certain class of fractional order integro-differential equations. Hacettepe Journal of Mathematics and Statistics, 46(2), 199-207.
AMA Aslıyüce S, Güvenilir AF, Zafer A. Oscillation criteria for a certain class of fractional order integro-differential equations. Hacettepe Journal of Mathematics and Statistics. April 2017;46(2):199-207.
Chicago Aslıyüce, Serkan, A. Feza Güvenilir, and Ağacık Zafer. “Oscillation Criteria for a Certain Class of Fractional Order Integro-differential Equations”. Hacettepe Journal of Mathematics and Statistics 46, no. 2 (April 2017): 199-207.
EndNote Aslıyüce S, Güvenilir AF, Zafer A (April 1, 2017) Oscillation criteria for a certain class of fractional order integro-differential equations. Hacettepe Journal of Mathematics and Statistics 46 2 199–207.
IEEE S. Aslıyüce, A. F. Güvenilir, and A. Zafer, “Oscillation criteria for a certain class of fractional order integro-differential equations”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 2, pp. 199–207, 2017.
ISNAD Aslıyüce, Serkan et al. “Oscillation Criteria for a Certain Class of Fractional Order Integro-differential Equations”. Hacettepe Journal of Mathematics and Statistics 46/2 (April 2017), 199-207.
JAMA Aslıyüce S, Güvenilir AF, Zafer A. Oscillation criteria for a certain class of fractional order integro-differential equations. Hacettepe Journal of Mathematics and Statistics. 2017;46:199–207.
MLA Aslıyüce, Serkan et al. “Oscillation Criteria for a Certain Class of Fractional Order Integro-differential Equations”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 2, 2017, pp. 199-07.
Vancouver Aslıyüce S, Güvenilir AF, Zafer A. Oscillation criteria for a certain class of fractional order integro-differential equations. Hacettepe Journal of Mathematics and Statistics. 2017;46(2):199-207.