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

2021/03/04 by Christopher S. Goodrich, Benjamin Lyons, Andrea Scapellato +1
Mathematics · #Functional Equations Stability Results #Mathematical Inequalities and Applications #Nonlinear Differential Equations Analysis

paper · doi:10.1080/10236198.2021.1894142

openalex publication_date 2021/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

We investigate relationships between the sign of the discrete fractional sequential difference (Δ1+a−μνΔaμf)(t) and the convexity of the function t↦f(t). In particular, we consider the case in which the bound (Δ1+a−μνΔaμf)(t)≥εf(a),for some ε>0 and where f(a)<0, is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation.

Citations

• Two monotonicity results for nabla and delta fractional differences
• Convexity for nabla and delta fractional differences
• Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one
• Monotonicity and convexity for nabla fractional (q, h)-differences
• A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity