Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

Authors

Christopher S. Goodrich, University of Nebraska at OmahaFollow
Benjamin Lyons, Creighton Preparatory School
Mihaela T. Velcsov, University of Nebraska at OmahaFollow

Document Type

Article

Publication Date

1-2021

Publication Title

Communications on Pure and Applied Analysis

Volume

20

Issue

1

First Page

339

Last Page

358

DOI

10.3934/cpaa.2020269

Abstract

We investigate the relationship between the sign of the discrete fractional sequential difference(Δ^v_1+a-μ Δ_a^μf)(t) and the monotonicity of the function t→f(t). More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound (Δ^v_1+a-μ Δ_a^μf)(t) ≥ -εf(a), for some ε >0. We prove that even though the fractional difference can be negative, the monotonicity of the function f, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation.

Comments

“This article has been published in a revised form in Communications on Pure and Applied Analysis (CPAA) http://dx.doi.org/10.3934/cpaa.2020269. This version is free to download for private research and study only. Not for redistribution, re-sale or use in derivative works.”

Recommended Citation

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure and Applied Analysis, 2021, 20(1): 339-358. doi: 10.3934/cpaa.2020269