Oscillation of Caputo Fractional Difference Equations with Damping Term

Year 2021, Volume: 21 Issue: 1, 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.

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