Analytical and numerical convexity results for discrete fractional sequential differences with negative lower bound. Issue 3 (4th March 2021)

• Record Type: Journal Article
• Authors: Goodrich, Christopher S.; Lyons, Benjamin; Scapellato, Andrea; Velcsov, Mihaela T.
• Is Part Of: Journal of difference equations and applications. Volume 27:Issue 3(2021)
• Page Start: 317
• Page End: 341
• Publication Date: 2021-03-04
• Subjects: Convexity -- discrete fractional calculus -- Gamma function -- sequential difference -- numerical approximation; Primary: 26A51 -- 33F05 -- 39A12 -- 65D15 -- 65Q20 -- Secondary: 39A99 -- 39B62; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2021.1894142
• Languages: English
• ISSNs: 1023-6198

Abstract

Abstract : We investigate relationships between the sign of the discrete fractional sequential difference ( Δ 1 + a − μ ν Δ a μ f ) ( t ) and the convexity of the function t ↦ f ( t ) . In particular, we consider the case in which the bound ( Δ 1 + a − μ ν Δ a μ f ) ( t ) ≥ ε f ( 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.