Opuscula Math. 40, no. 2 (2020), 227-239
https://doi.org/10.7494/OpMath.2020.40.2.227
Opuscula Mathematica
On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms
John R. Graef
Said R. Grace
Ercan Tunç
Abstract. This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.
Keywords: integro-differential equations, fractional differential equations, nonoscillatory solutions, boundedness, Caputo derivative.
Mathematics Subject Classification: 34E10, 34A34.
- D. Bǎleanu, J.A.T. Machado, A.C.J. Luo, Fractional Dynamics and Control, Springer, New York, 2012.
- M. Bohner, S.R. Grace, N. Sultana, Asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations, Opuscula Math. 34 (2014), 5-14.
- E. Brestovanská, M. Medveď, Asymptotic behavior of solutions to second-order differential equations with fractional derivative perturbations, Electron. J. Differ. Eq. 2014 (2014), 1-10.
- M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. Royal Astronom. Soc. 13 (1967), 529-535.
- K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010.
- K.M. Furati, N.-E. Tatar, Power-type estimates for a nonlinear fractional differential equations, Nonlinear Anal. 62 (2005), 1025-1036.
- S.R. Grace, A. Zafer, Oscillatory behavior of integro-dynamic and integral equations on time scales, Appl. Math. Lett. 28 (2014), 47-52.
- S.R. Grace, A. Zafer, On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations, Eur. Phys. J. Special Topics 226 (2017), 3657-3665.
- S.R. Grace, R.P. Agarwal, P.J.Y. Wong, A. Zafer, On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal. 15 (2012), 222-231.
- S.R. Grace, J.R. Graef, S. Panigrahi, E. Tunҫ, On the oscillatory behavior of Volterra integral equations on time-scales, PanAmerican Math. J. 23 (2013), 35-41.
- S.R. Grace, J.R. Graef, E. Tunҫ, On the boundedness of nonoscillatory solutions of certain fractional differential equations with positive and negative terms, Appl. Math. Lett. 97 (2019), 114-120.
- S.R. Grace, J.R. Graef, A. Zafer, Oscillation of integro-dynamic equations on time scales, Appl. Math. Lett. 26 (2013), 383-386.
- J.R. Graef, L. Kong, A. Ledoan, M. Wang, Modeling online social network dynamics using fractional-order epidemiological models, to appear.
- G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1988, Reprint of the 1952 edition.
- A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, 2006.
- V. Lakshmikantham, S. Leela, J. Vaaundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
- Q.-H. Ma, J. Pečarić, 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), 3258-3267.
- M. Medveď, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math. Anal. Appl. 214 (1997), 349-366.
- M. Medveď, Integral inequalities and global solutions of semilinear evolution equations, J. Math. Anal. Appl. 267 (2002), 643-650.
- M. Medveď, Asymptotic integration of some classes of fractional differential equations, Tatra Mt. Math. Publ. 54 (2013), 119-132.
- M. Medveď, M. Pospíšil, Asymptotic integration of fractional differential equations with integrodifferential right-hand side, Math. Modelling Anal. 20 (2015), 471-489.
- K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York, 1993.
- I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego, 1999.
- A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integral and Series: Elementary Functions, vol. 1, Nauka, Moscow, 1981 [in Russian].
- S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993.
- John R. Graef (corresponding author)
https://orcid.org/0000-0002-8149-4633- University of Tennessee at Chattanooga, Department of Mathematics, Chattanooga, TN 37403, USA
- Said R. Grace
- https://orcid.org/0000-0001-8783-5227
- Cairo University, Faculty of Engineering, Department of Engineering Mathematics, Orman, Giza 12221, Egypt
- Ercan Tunç
- https://orcid.org/0000-0001-8860-608X
- Gaziosmanpasa University, Department of Mathematics, Faculty of Arts and Sciences, 60240, Tokat, Turkey
- Communicated by Josef Diblík.
- Received: 2019-08-06.
- Revised: 2019-10-09.
- Accepted: 2019-11-03.
- Published online: 2020-03-09.
