Oscillation of Caputo Fractional Difference Equations with Damping Term

Year 2021, , 106 - 112, 24.02.2021

Tuğba Yalçın Uzun Sermin Öztürk Hüsniye Öz

https://doi.org/10.35414/akufemubid.803511

Abstract

In this paper, we obtain a sufficent condition for the oscillation of forced fractional difference equations with damping term of the form (1+p(t))∆(∆_C^α x(t))+p(t) ∆_C^α x(t)+f(t,x(t))=g(t),t∈N_0
with initial condition ∆^k ├ x(t)┤|_(t=0)=x_k,k=1,2,…,n-1 where α∈(n-1,n) is a constant (n∈N), ∆_C^α x is the Caputo fractional difference operator of order α of x and N_0={0,1,2,…}. For this study, the proposition “p(t) and g(t) are real functions, p(t)>-1,f:N_0×R⟶R and x≠0,t_0∈N_0” is held. An illustrative example is given at the end of the paper.

Keywords

Oscillation, Forced fractional difference equation, Damping term, Caputo difference operator

Project Number

18.FEN.BİL.67

Sönüm Terimli Caputo Kesirli Fark Denklemlerinin Salınımlılığı

Abstract

Bu makalede, α∈(n-1,n) bir sabit (n∈〖N) ∆〗_C^α x, x’in α-yıncı mertebeden kesirli Caputo kesirli fark operatörü ve N_0={0,1,2,…} olmak üzere, ∆^k ├ x(t)┤|_(t=0)=x_k,k=1,2,…,n-1 başlangıç şartına sahip
(1+p(t))∆(∆_C^α x(t))+p(t) ∆_C^α x(t)+f(t,x(t))=g(t),t∈N_0 ile verilen ikinci taraflı sönüm terimli kesirli fark denkleminin salınımlılığı için bir yeter şart elde edilmiştir. Bu çalışma için “p(t) ve g(t) reel fonksiyonlar, p(t)>-1,f:N_0×R⟶R ve x≠0,t_0∈N_0” önermesi geçerlidir. Makalenin sonunda açıklayıcı bir örnek verilmiştir.

Keywords

Salınımlılık, İkinci taraflı kesirli fark denklemi, sönüm terimi, Caputo fark operatörü

Supporting Institution

Afyon Kocatepe Üniversitesi Bilimsel Araştırma Projeleri Koordinasyon Birimi

Project Number

18.FEN.BİL.67

Thanks

Bu çalışma, Afyon Kocatepe Üniversitesi Bilimsel Araştırma Projeleri Koordinasyon Birimi tarafından 18.FEN.BİL.67 numaralı proje ile desteklenmiştir. Ayrıca, yapıcı yorumları ve katkılarından dolayı değerli hakemlere teşekkürü bir borç biliriz.

References

• Abdalla, B., Abodayeh, K., Abdeljawad, Th., Alzabut, J., 2017. New oscillation criteria for forced nonlinear fractional difference equations. Vietnam Journal of Mathematics, 45, 609¬¬–618.
• Abdalla, B., Alzabut, J., Abdeljawad, T., 2018. On the oscillation of higher order fractional difference equations with mixed nonlinearities. Hacettepe Journal of Mathematics and Statistics, 47, 207–217.
• Abdeljawad, T., 2011. On Riemann and Caputo fractional differences. Computers & Mathematics with Applications, 62, 1602–1611.
• Alzabut, J.O., Abdeljawad, T., 2014. Sufficient conditions for the oscillation of nonlinear fractional difference equations. Journal of Fractional Calculus and Applications, 5, 177¬–187.
• Chen, D., Qu, P., Lan, Y., 2013. Forced oscillation of certain fractional differential equations. Advances in Difference Equations, 125, 1–10. Chen, D.X., 2013. Oscillatory behavior of a class of fractional differential equations with damping. University Politehnica of Bucharest Scientific Bulletin, 75, 107–118.
• Grace, S.R., Agarwal, R.P., Wong, P.J.Y., Zafer, A., 2012. On the oscillation of fractional differential equations. Fractional Calculus and Applied Analysis, 15, 222–231.
• Li, W.N., 2015. Forced oscillation criteria for a class of fractional partial differential equations with damping term. Mathematical Problems in Engineering, 2015, 1–6.
• Li, W.N., 2016. Oscillation results for certain forced fractional difference equations with damping term. Advances in Difference Equations, 70, 1–9. Sagayaraj, M.R., Selvam, A.G.M., Loganathan, M.P., 2014. On the oscillation of nonlinear fractional difference equations. Mathematica Aeterna, 4, 91–99. Tunc, E., Tunc, O. 2016. On the oscillation of a class of damped fractional differential equations. Miskolc Mathematical Notes, 17, 647–656.
• Yang, J., Liu, A., Liu, T., 2015. Forced oscillation of nonlinear fractional differential equations with damping term. Advances in Difference Equations, 1, 1–7.