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

Author:

Arundhathi Sivakumar1,Alzabut Jehad23,Muthulakshmi Velu1,Adıgüzel Hakan4

Affiliation:

1. Department of Mathematics , Periyar University , Salem-636 011, Tamilnadu , India

2. Department of Mathematics and Sciences , Prince Sultan University , 11586 Riyadh , Saudi Arabia

3. Department of Industrial Engineering , OSTİM Technical University , Ankara 06374 , Turkey

4. Department of Engineering Fundamental Sciences , Sakarya University of Applied Sciences , 54580 Sakarya , Turkey

Abstract

Abstract In this study, we have investigated the oscillatory properties of the following fractional difference equation: α + 1 χ ( κ ) α χ ( κ ) p ( κ ) г ( α χ ( 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 N 0 \kappa \in {{\mathbb{N}}}_{0} , α {\nabla }^{\alpha } denotes the Liouville fractional difference operator of order α ( 0 , 1 ) \alpha \in \left(0,1) , p p , and q q are nonnegative sequences, and г г 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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

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