×

Two monotonicity results for nabla and delta fractional differences. (English) Zbl 1327.39011

The authors establish two monotonicity results for the nabla and delta fractional differences (Theorems A and B), which emphasize a nontrivial differences between these two concepts. Moreover, Theorem B implies that Corollary 2.3 in the paper by R. Dahal and Ch. S. Goodrich [Arch. Math. 102, No. 3, 293–299 (2014; Zbl 1296.39016)] is incorrect without an additional assumption, cf. the erratum of R. Dahal and Ch. S. Goodrich [ibid. 104, No. 6, Article ID 771, 599–600 (2015; Zbl 1330.39022)]. This fact is also illustrated by a counterexample.

MSC:

39A70 Difference operators
39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anastassiou G.: Foundations of nabla fractional calculus on time scales and inequalities. Comput. Math. Appl. 59, 3750-3762 (2010) · Zbl 1198.26033 · doi:10.1016/j.camwa.2010.03.072
[2] F. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), 981-989. · Zbl 1166.39005
[3] F. Atici and P. Eloe, Discrete fractional calculus with the nabla operator, Elect. J. Qual. Theory Differential Equations, Spec. Ed. I No. 3 (2009), 1-12. · Zbl 1189.39004
[4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications Birkhäuser, Boston (2001). · Zbl 0978.39001
[5] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston (2003). · Zbl 1025.34001
[6] R. Dahal and C. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math. 102 (2014), 293-299. · Zbl 1296.39016
[7] R. A. C. Ferreira, A discrete fractional Gronwall inequality, Proc. Amer. Math. Soc. 140 (2012), 1605-1612. · Zbl 1243.26012
[8] R. A. C. Ferreira and D. F. M. Torres, Fractional h-difference equations arising from the calculus of variations, Appl. Anal. Discrete Math. 5 (2011), 110-121. · Zbl 1289.39007
[9] C. Goodrich and A. Peterson, Discrete Fractional Calculus, Springer, Preliminary Version, 2014. · Zbl 1350.39001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.