Oscillation Theorems for Certain Forced Nonlinear Discrete Fractional Order Equations

George E. Chatzarakis, A. George Maria Selvam, R. Janagaraj, Maria Douka

Abstract


The main objective of this work is to obtain some new sufficient conditions that are essential for the oscillation of the solutions of forced nonlinear discrete fractional equations of the form
\begin{align*}
\Delta\left[\Delta^\mu(u(j))\right]+\eta(j)\Phi(u(j))=\psi(j), \ \ j\in N_0
\end{align*}
where \(\Delta^{\mu-1}u(0)=u_0\); \(\Delta u(j)=u(j+1)-u(j)\) and \(\Delta^\mu\) is defined as the difference operator of the Riemann-Liouville (R-L) derivative of order \(\mu\in(0,1]\) and \(N_0=\{0,1,2,\cdots\}\). Numerical examples are presented to show the validity of the theoretical results.


Keywords


Oscillation; Fractional order difference equations; Forcing term

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References


T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl. 62(3) (2011), 1602 – 1611, DOI: 10.1016/j.camwa.2011.03.036.

J. O. Alzabut and T. Abdeljawad, Sufficient conditions for the oscillation of nonlinear fractional difference equations, Journal of Fractional Calculus and Applications 5(1) (2014), 177 – 187.

G. A. Anastassiou, About discrete fractional calculus with inequalities, in Intelligent Mathematics: Computational Analysis, Intelligent Systems Reference Library, Vol. 5, Springer, Berlin — Heidelberg, DOI: 10.1007/978-3-642-17098-0_35.

F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ. 2(2) (2007), 165 – 176.

F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137(3) (2009), 981 – 989, DOI: 10.1090/S0002-9939-08-09626-3.

G. E. Chatzarakis, P. Gokulraj and T. Kalaimani, Oscillation test for fractional difference equations, Tatra Mt. Math. Publ. 71 (2018), 53 – 64, DOI: 10.2478/tmmp-2018-0005.

G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam, Oscillatory solutions of nonlinear fractional difference equations, Int. J. Difference Equ. 13(1) (2018), 19 – 31.

J. B. Diaz and T. J. Osler, Differences of fractional order, Math. Comp. 28(125) (1974), 185 – 202, DOI: 10.2307/2005825.

S. Elaydi, An Introduction to Difference Equations, 3-e, Springer International Edition (2005), DOI: 10.1007/978-1-4757-9168-6.

Q. Feng and F. Meng, Oscillation of solutions to nonlinear forced fractional differential equations, Electron. J. Differential Equations 2013(169) (2013), 1 – 10.

A. George Maria Selvam and R. Janagaraj, Oscillation theorems for damped fractional order difference equations, AIP Conference Proceedings 2095(030007) (2019), 1 – 7, DOI: 10.1063/1.5097518.

A. George Maria Selvam and R. Janagaraj, Oscillatory behavior of fractional order difference equations with damping, American International Journal of Research in Science, Technology, Engineering & Mathematics 19-06 (2019), 34 – 41.

A. George Maria Selvam and R. Janagaraj, Oscillation criteria of a class of fractional order damped difference equations, Int. J. Appl. Math. 32(3) (2019), 433 – 441, DOI: 10.12732/ijam.v32i3.5.

A. George Maria Selvam, M. P. Loganathan, R. Janagaraj and D. A. Vianny, Oscillation of fractional difference equations with damping terms, JP Journal of Mathematical Sciences 16(1) (2016), 1 – 13.

C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer International Publishing, Switzerland (2015), DOI: 10.1007/978-3-319-25562-0.

S. R. Grace, R. P. Agarwal, P. J. Y. Wong, A. Zafer, On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal. 15(2) (2012), 222 – 231, DOI: 10.2478/s13540-012-0016-1.

H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp. 50(182) (1988), 513 – 529, DOI: 10.1090/S0025-5718-1988-0929549-2.

S. Harikrishnan, P. Prakash and J. J. Nieto, Forced oscillation of solutions of a nonlinear fractional partial differential equations, Appl. Math. Comput. 254 (2015), 14 – 19, DOI: 10.1016/j.amc.2014.12.074.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 1-e, North-Holland Mathematics Studies, 204 (2006).

B. Kuttner, On differences of fractional order, Proc. Lond. Math. Soc. 3-7(1) (1957), 453 – 466, DOI: 10.1112/plms/s3-7.1.453.

W. Li, W. Sheng and P. Zhang, Oscillatory properties of certain nonlinear fractional nabla difference equations, Journal of Applied Analysis and Computation 8(6) (2016), 1910 – 1918, DOI: 10.11948/2018.1910.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, USA (1993).

V. Muthulakshmi and S. Pavithra, Oscillatory behavior of fractional differential equations with damping, International Journal of Mathematics and its Applications 4(4-c) (2017), 383 – 388.

A. Secer and H. Adiguzel, Oscillation of solutions for a class of nonlinear fractional difference equations, J. Nonlinear Sci. Appl. 9 (2016), 5862 – 5869, DOI: 10.22436/jnsa.009.11.14.

J. Yang, A. Liu and T. Liu, Forced oscillation of nonlinear fractional difference equations with damping term, Adv. Difference Equ. 2015(1) (2015), 1 – 7, DOI: 10.1186/s13662-014-0331-4.




DOI: http://dx.doi.org/10.26713%2Fcma.v10i4.1286

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