Oscillation of Caputo Fractional Difference Equations with Damping Term

Objective & Hypothesis: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 𝛼𝑥(𝑡)) + 𝑝(𝑡)∆𝐶 𝛼𝑥(𝑡) + 𝑓(𝑡, 𝑥(𝑡)) = 𝑔(𝑡), 𝑡 ∈ ℕ0with initial condition ∆ 𝑘𝑥(𝑡)|𝑡=0 = 𝑥𝑘, 𝑘 = 1,2,… , 𝑛 − 1 where 𝛼 ∈ (𝑛 − 1, 𝑛) is a constant (𝑛 ∈ ℕ), ∆𝐶 𝛼𝑥 is the Caputo fractional difference operator of order 𝛼 of 𝑥 and ℕ0 = {0,1,2,… }.

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Key Findings & Data:For this study, the proposition “𝑝(𝑡) and 𝑔(𝑡) are real functions, 𝑝(𝑡) > −1, 𝑓: ℕ0 × ℝ ⟶ ℝ and 𝑥 ≠ 0, 𝑡0 ∈ ℕ0 ” is held.

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Conclusion & Impact:An illustrative example is given at the end of the paper.

🔬Design:Research Article

📍Scope:Comprehensive Sample

⚙️Variables:Domain Parameters

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 𝛼𝑥(𝑡)) + 𝑝(𝑡)∆𝐶 𝛼𝑥(𝑡) + 𝑓(𝑡, 𝑥(𝑡)) = 𝑔(𝑡), 𝑡 ∈ ℕ0with initial condition ∆ 𝑘𝑥(𝑡)|𝑡=0 = 𝑥𝑘, 𝑘 = 1,2,… , 𝑛 − 1 where 𝛼 ∈ (𝑛 − 1, 𝑛) is a constant (𝑛 ∈ ℕ), ∆𝐶 𝛼𝑥 is the Caputo fractional difference operator of order 𝛼 of 𝑥 and ℕ0 = {0,1,2,… }. For this study, the proposition “𝑝(𝑡) and 𝑔(𝑡) are real functions, 𝑝(𝑡) > −1, 𝑓: ℕ0 × ℝ ⟶ ℝ and 𝑥 ≠ 0, 𝑡0 ∈ ℕ0 ” is held. An illustrative example is given at the end of the paper.

Özet

Bu makalede, 𝛼 ∈ (𝑛 − 1, 𝑛) bir sabit (𝑛 ∈ ℕ) ∆𝐶𝛼𝑥, 𝑥’in 𝛼-yıncı mertebeden kesirli Caputo kesirli farkoperatörü ve ℕ0 = {0,1,2,… } olmak üzere, ∆𝑘𝑥(𝑡)|𝑡=0 = 𝑥𝑘,𝑘 = 1,2,…, 𝑛 − 1 başlangıç şartına sahip(1 + 𝑝(𝑡))∆(∆𝐶𝛼𝑥(𝑡)) + 𝑝(𝑡)∆𝐶𝛼𝑥(𝑡) + 𝑓(𝑡, 𝑥(𝑡)) = 𝑔(𝑡), 𝑡 ∈ ℕ0ile verilen ikinci taraflı sönüm terimlikesirli fark denkleminin salınımlılığı için bir yeter şart elde edilmiştir. Bu çalışma için “𝑝(𝑡) ve 𝑔(𝑡) reelfonksiyonlar, 𝑝(𝑡) > −1, 𝑓: ℕ0 × ℝ ⟶ ℝ ve 𝑥 ≠ 0, 𝑡0 ∈ ℕ0” önermesi geçerlidir. Makalenin sonundaaçıklayıcı bir örnek verilmiştir.

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Cite This Article

UZUN, T. Y., ÖZTÜRK, S., ÖZ, H. (2022). Oscillation of Caputo Fractional Difference Equations with Damping Term. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, Cilt: 21 - Sayı: 1, 106–112. https://ajindex.com/articles/429645

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