×

Convexity for nabla and delta fractional differences. (English) Zbl 1320.39003

If \((a,b)\in \mathbb R^2\) and \(b>a\), denote \[ N_a= \{a,a+1,a+2,\dots\},\;N^b_a= \{a,a+1, a+2,\dots, b\}. \] The authors prove results for nabla and delta fractional differences:
(1) If \(f: N_{a+1}\to \mathbb R\) satisfies \(\nabla^\nu_a f(t)\geq 0\) for each \(t\in N_{a+1}\), with \(2\nu<3\), then \(\nabla^2 f(t)\geq 0\) for \(t\in N_{a+3}\).
(2) If \(f: N_a\to \mathbb R\) satisfies \(\nabla^\nu_a f(t)\geq 0\) for each \(t\in N_a\), with \(2<\nu<3\), and \(f(a)\leq 0\), \(\Delta f(a)\geq 0\), \(\Delta^2 f(a)\geq 0\), then \(\Delta f(t)\geq 0\) for \(t\in N_a\).
The authors consider the general case \(N-1<\nu<N\) with \(N>3\) and prove:
(3) \(\Delta^\nu_a f(t)\geq 0\) for each \(t\in N_{a+1}\), with \(N- 1<\nu< N\), \(N\in N_1\), \((-1)^{N-i}\Delta^i f(a)\geq 0\), \(0\leq i\leq N-2\) and \(\Delta^{N-1}f(a)\geq 0\), then \(\Delta^{N-1} f(t)\geq 0\) for \(t\in N_a\).

MSC:

39A12 Discrete version of topics in analysis
39A70 Difference operators
26B12 Calculus of vector functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1090/S0002-9939-08-09626-3 · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[2] Atici F.M., Electron. J. Qual. Theory Differ. Equ. 3 pp 1– (2009) · Zbl 1189.39004 · doi:10.14232/ejqtde.2009.4.3
[3] DOI: 10.1016/j.camwa.2010.03.072 · Zbl 1198.26033 · doi:10.1016/j.camwa.2010.03.072
[4] J.Baoguo, L.Erbe and A.Peterson, Two monotonicity results for nabla and delta fractional differences, submitted for publication. · Zbl 1327.39011
[5] DOI: 10.1007/978-1-4612-0201-1 · doi:10.1007/978-1-4612-0201-1
[6] DOI: 10.1007/978-0-8176-8230-9 · doi:10.1007/978-0-8176-8230-9
[7] DOI: 10.1007/s00013-014-0620-x · Zbl 1296.39016 · doi:10.1007/s00013-014-0620-x
[8] DOI: 10.1016/j.camwa.2009.05.012 · Zbl 1189.26044 · doi:10.1016/j.camwa.2009.05.012
[9] DOI: 10.1090/S0002-9939-2012-11533-3 · Zbl 1243.26012 · doi:10.1090/S0002-9939-2012-11533-3
[10] C.Goodrich and A.Peterson, Discrete Fractional Calculus, Preliminary Version,2014.
[11] DOI: 10.1016/j.aml.2014.04.013 · Zbl 1314.26010 · doi:10.1016/j.aml.2014.04.013
[12] W.Kelley and A.Peterson, Difference Equations: An Introduction with Applications, 2nd ed., Academic Press, Harcourt, 2001. · Zbl 0970.39001
[13] M.Holm, The theory of discrete fractional calculus: Development and applications, Ph.D. diss.,University of Nebraska-Lincoln,2011.
[14] DOI: 10.4067/S0719-06462011000300009 · Zbl 1248.39003 · doi:10.4067/S0719-06462011000300009
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.