# ON THE OSCILLATION OF THREE DIMENSIONAL alpha-FRACTIONAL DIFFERENTIAL SYSTEMS

 Title ON THE OSCILLATION OF THREE DIMENSIONAL alpha-FRACTIONAL DIFFERENTIAL SYSTEMS Publication Type Journal Article Year of Publication 2018 Authors CHATZARAKIS G.E., DEEPA M., NAGAJOTHI N., SADHASIVAM V. Journal Dynamic Systems and Applications Volume 27 Issue 4 Start Page 873 Pagination 22 Date Published 11/2018 ISSN 1056-2176 AMS Subject Classification 34A08, 34A34, 34K11 Abstract In this article, we consider the three dimensional $\alpha$-fractional nonlinear differential system of the form $$D^{\alpha}\left(x(t)\right)=a(t)f\left(y(t)\right),$$ $$D^{\alpha}\left(y(t)\right)=-b(t)g\left(z(t)\right),$$ $$D^{\alpha}\left(z(t)\right)=c(t)h\left(x(t)\right),\quad t \geq t_0,$$ where $0 < \alpha \leq 1$, $D^{\alpha}$ denotes the Katugampola fractional derivative of order $\alpha$. We establish some new sufficient conditions for the oscillation of the solutions of the differential system, using the generalized Riccati transformation and inequality technique. Examples illustrating the results are also given. PDF https://acadsol.eu/dsa/articles/27/4/12.pdf DOI 10.12732/dsa.v27i4.12 Refereed Designation Refereed Full Text [1] S. Abbas, M. Benchohra and G.M. N’Guerekata, Topics in fractional differential equations, Springer, Newyork, 2012. [2] T. Abdeljawad, On conformable fractional calculus, J. Compu. Appl. Math., 279 (2015) 57-66. [3] A. Atangana, D. Baleanu and A. Alsaedi, New properties of conformable deriva- tive, Open Math., 13 (2015) 889-898. [4] D.R. Anderson and D.J. Ulnessn, Properties of the Katugampola fractional derivative with potential application in quantum mechanics, J. Math. Phy., 56 (6), 063502 (2015). [5] G.E. Chatzarakis, S.R. Grace and I. Jadlovska, Oscillation criteria for third-order delay differential equations, Adv. Difference Equ., 2017:330. [6] V. Daftardar-Gejji, Fractional calculus theory and applications, Narosa publish- ing house Pvt. Ltd., 2014. [7] K. Diethelm, The analysis of fractional differential equations, Springer, Berlin, 2010. [8] M. Dosoudilova, A. Lomtatidze and J. Sremr, Oscillatory properties of solutions to two-dimensional Emden-Fowler type equations, International workshop qual- itde, Georgia, (2014) 18-20. [9] L.H. Erbe, Q.K. Kong and B.G. Zhang, Oscillation theory for functional differ- ential equations, Marcel Dekker, New York, 1995. [10] R. Hilfer, Applications of fractional calculus in physics, World scientific pub. co., 2000. [11] H.F. Huo andW.T. Li, Oscillation criteria for certain two-dimensional differential systems, International journal of applied mathematics, 6 (3) (2001) 253-261. [12] U.N. Katugampola, A new fractional derivative with classical properties, e-print arXiv:1410.6535, (2014). [13] U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6(4) (2014) 1-15. [14] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of frac- tional derivatives, J. Compu. Appl. Math., 264 (2014) 65-70. [15] I.T. Kiguradze and T.A. Chaturia, Asymptotic properties of solutions of nonau- tonomous ordinary differential equations, Kluwer Acad. Publ., Dordrecht, 1993. [16] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of frac- tional differential equations, Elsevier science, B.V. Amsterdam, 2006. [17] I-G.E. Kordonis and Ch.G. Philos, On the oscillation of nonlinear two-dimensional differential systems, Proc. Amer. Math. Soc., 126 (1998) 1661-1667. [18] W.T. Li and S.S. Cheng, Limiting behaviours of non-oscillatory solutions of a pair of coupled nonlinear differential equations, Proc. of the Edinburgh Math. Soc., 43 (2000) 457-473. [19] C. Milici and G. Draganescu, Introduction to fractional calculus, Lambert aca- demic publishing, 2016. [20] A.K. Nandakumaran, P.S. Datti and Raju K. George, Ordinary differential equa- tions, principles and applications, Cambridge University press, 2017. [21] S. Padhi and S. Pati, Theory of third order differential equations, Electronic journal of differential equations, Springer, New Delhi, 2014. [22] I. Podlubny, Fractional differential equations, Academic press, San Diego, Calif, USA, 1999. [23] S. Harikrishnan, P. Prakash, and J.J. Nieto, Forced oscillation of solutions of a nonlinear fractional partial differential equation, Applied Mathematics and com- putation, 254 (2015) 14-19. [24] B. Rai, H.I. Freedman and J.F. Addicott, Analysis of three spieces model of mutualism in predator-prey and competitive systems, Mathematical biosciences, 65 (1983) 13-50. [25] H.H. Robertson, The solution of a set of reaction rate equations, Academic press, (1966) 178-182. [26] V. Sadhasivam, J. Kavitha andM. Deepa, Existence of solutions of three-dimensional fractional differential systems, Applied mathematics, 8 (2017) 193-208. [27] V. Sadhasivam, M. Deepa and N. Nagajothi, On the oscillation of nonlinear frac- tional differential systems, International journal of advanced research in science, engineering and technology, 4 (11) (2017) 4861-4867. [28] S.H. Saker, Oscillation theory of delay differential and difference equations, VDM Verlag Dr.Muller Aktiengesellschaft and Co, USA, 2010. [29] H. Samajova, B. Ftorek and E. Spanikova, Asymptotic character of non-oscillatory solutions to functional differential systems, Electronic journal of qualitative the- ory of differential equations, 42 (2015) 1-13. [30] S.G. Samko, A.A. Kilbas and O.I. Mirchev, Fractional integrals and derivatives, Theory and applications, Gordon and breach science publishers, Singapore, 1993. [31] B.V. Shirokorad, On the existence of a cycle provided that conditions for abso- lute stability of a three-dimensional system are not fulfilled, Avtomatika i Tele- mekhanika, 19 (1958) 953-967. [32] E. Spanikova and Zilina, Oscillatory properties of solutions of three-dimensional differential systems of neutral type, Czechoslovak mathematical journal, 50 (125) (2000) 879-887. [33] Vasily E. Tarasov, Fractional dynamics, Springer, 2009. [34] P. Weng and Z. Xu, On the oscillation of a two-dimensional differential system, International journal of qualitative theory of differential equations and applica- tions, 2 (1) (2008) 38-46. [35] Y. Zhou, Basic theory of fractional differential equations, World scientific, Sin- gapore, 2014. 894