<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this note, we discuss the definitions of the difference sequences defined earlier by Kızmaz (1981), Et and Çolak (1995), Malkowsky et al. (2007), Başar (2012), Baliarsingh (2013, 2015) and many others. Several authors have defined the difference sequence spaces and studied their various properties. It is quite natural to analyze the convergence of the corresponding sequences. As a part of this work, a convergence analysis of difference sequence of fractional order defined earlier is presented. It is demonstrated that the convergence of the fractional difference sequence is dynamic in nature and some of the results involved are also inconsistent. We provide certain stronger conditions on the primary sequence and the results due to earlier authors are substantially modified. Some illustrative examples are provided for each point of the modifications. Results on certain operator norms related to the difference operator of fractional order are also determined.</p></abstract>

In this note, we discuss the definitions of the difference sequences defined earlier by Kızmaz (1981), Et and Çolak (1995), Malkowsky et al. (2007), Başar (2012), Baliarsingh (2013, 2015) and many others. Several authors have defined the difference sequence spaces and studied their various properties. It is quite natural to analyze the convergence of the corresponding sequences. As a part of this work, a convergence analysis of difference sequence of fractional order defined earlier is presented. It is demonstrated that the convergence of the fractional difference sequence is dynamic in nature and some of the results involved are also inconsistent. We provide certain stronger conditions on the primary sequence and the results due to earlier authors are substantially modified. Some illustrative examples are provided for each point of the modifications. Results on certain operator norms related to the difference operator of fractional order are also determined.

On the convergence difference sequences and the related operator norms

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"On the convergence difference sequences and the related operator norms"}

In this note, we discuss the definitions of the difference sequences defined earlier by Kızmaz (1981), Et and Çolak (1995), Malkowsky et al. (2007), Başar...

On the convergence difference sequences and the related operator norms

Published Online: Dec 02, 2020

Page range: 245 - 259

Received: Mar 05, 2020

DOI: https://doi.org/10.2478/ausm-2020-0016

Keywords
sequence spaces, difference sequence spaces, difference operators Δ and Δ, operator norms

© 2020 P. Baliarsingh et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On the convergence difference sequences and the related operator norms

Published Online: Dec 02, 2020

Page range: 245 - 259

Received: Mar 05, 2020

DOI: https://doi.org/10.2478/ausm-2020-0016

Keywordssequence spaces, difference sequence spaces, difference operators Δ and Δ, operator norms

© 2020 P. Baliarsingh et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
sequence spaces, difference sequence spaces, difference operators Δ and Δ, operator norms