Oscillation results for certain forced fractional difference equations with damping term

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• 作者：Wei Nian Li
• 关键词：oscillation ; forced fractional difference equation ; damping term
• 刊名：Advances in Difference Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,440 KB
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847

文摘

In this paper, we establish two sufficient conditions for the oscillation of forced fractional difference equations with damping term of the form $$\bigl(1+p(t)\bigr)\Delta\bigl(\Delta^{\alpha}x(t)\bigr)+p(t) \Delta^{\alpha}x(t)+f\bigl(t,x(t)\bigr)=g(t),\quad t\in\mathbb{N}_{0}, $$ with initial condition \(\Delta^{\alpha-1}x(t)|_{t=0}=x_{0}\), where \(0<\alpha<1 \) is a constant, \(\Delta^{\alpha}x\) is the Riemann-Liouville fractional difference operator of order α of x, and \(\mathbb{N}_{0}=\{0,1,2,\ldots\}\). Keywords oscillation forced fractional difference equation damping term MSC 26A33 39A12 39A21 1 IntroductionIn the past few years, the theory of fractional differential equations and their applications have been investigated extensively. For example, see monographs [1–4]. In recent years, fractional difference equations, which are the discrete counterpart of the corresponding fractional differential equations, have comparably gained attention by some researchers. Many interesting results were established. For instance, see papers [5–20] and the references therein.