A certain class of fractional difference equations with damping: Oscillatory properties

Abstract

In this study, we have investigated the oscillatory properties of the following fractional difference equation: ∇α+1χ(κ)⋅∇αχ(κ)−p(κ)г(∇αχ(κ))+q(κ)G∑μ=κ−α+1∞(μ−κ−1)(−α)χ(μ)=0,{\nabla }^{\alpha +1}\chi \left(\kappa )\cdot {\nabla }^{\alpha }\chi \left(\kappa )-p\left(\kappa )г\left({\nabla }^{\alpha }\chi \left(\kappa ))+q\left(\kappa ){\mathcal{G}}\left(\mathop{\sum }\limits_{\mu =\kappa -\alpha +1}^{\infty }{\left(\mu -\kappa -1)}^{\left(-\alpha )}\chi \left(\mu )\right)=0, where κ∈N0\kappa \in {{\mathbb{N}}}_{0}, ∇α{\nabla }^{\alpha } denotes the Liouville fractional difference operator of order α∈(0,1)\alpha \in \left(0,1), pp, and qq are nonnegative sequences, and гг and G{\mathcal{G}} are real valued continuous functions, all of which satisfy certain assumptions. Using the generalized Riccati transformation technique, mathematical inequalities, and comparison results, we have found a number of new oscillation results. A few examples have been built up in this context to illustrate the main findings. The conclusion of this study is regarded as an expansion of continuous time to discrete time in fractional contexts.

Keywords

• oscillation
• fractional difference equation
• damping
• riccati transformation
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