A convexity result for fractional differences. (English) Zbl 1314.26010

Summary: In this brief note we demonstrate that under certain conditions the positivity of the fractional difference \(\varDelta_0^\mu y(t)\), for a given function \(y : \mathbb{N}_0 \to \mathbb{R}\) and a number \(\mu \in(2, 3)\), implies the convexity of \(y\). This is given as a special case of a more general result. Finally, as a concrete application of this result we demonstrate that a particular class of fractional boundary value problems has no nontrivial positive solutions.


26A33 Fractional derivatives and integrals
39A12 Discrete version of topics in analysis
Full Text: DOI


[1] Atici, F. M.; Eloe, P. W., A transform method in discrete fractional calculus, Int. J. Difference Equ., 2, 165-176 (2007)
[2] Atici, F. M.; Eloe, P. W., Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137, 981-989 (2009) · Zbl 1166.39005
[3] Atici, F. M.; Eloe, P. W., Two-point boundary value problems for finite fractional difference equations, J. Difference Equ. Appl., 17, 445-456 (2011) · Zbl 1215.39002
[4] Dahal, R.; Duncan, D.; Goodrich, C. S., Systems of semipositone discrete fractional boundary value problems, J. Difference Equ. Appl., 20, 473-491 (2014) · Zbl 1319.39002
[5] Ferreira, R. A.C., Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one, J. Difference Equ. Appl., 19, 712-718 (2013) · Zbl 1276.26013
[6] Ferreira, R. A.C.; Goodrich, C. S., Positive solution for a discrete fractional periodic boundary value problem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 19, 545-557 (2012) · Zbl 1268.26010
[7] Goodrich, C. S., Solutions to a discrete right-focal boundary value problem, Int. J. Difference Equ., 5, 195-216 (2010)
[8] Goodrich, C. S., On discrete sequential fractional boundary value problems, J. Math. Anal. Appl., 385, 111-124 (2012) · Zbl 1236.39008
[9] Goodrich, C. S., On a discrete fractional three-point boundary value problem, J. Difference Equ. Appl., 18, 397-415 (2012) · Zbl 1253.26010
[10] Goodrich, C. S., On a first-order semipositone discrete fractional boundary value problem, Arch. Math. (Basel), 99, 509-518 (2012) · Zbl 1263.26016
[11] Goodrich, C. S., On semipositone discrete fractional boundary value problems with nonlocal boundary conditions, J. Difference Equ. Appl., 19, 1758-1780 (2013) · Zbl 1282.26008
[12] Atici, F. M.; Eloe, P. W., Linear systems of fractional nabla difference equations, Rocky Mountain J. Math., 41, 353-370 (2011) · Zbl 1218.39003
[13] Atici, F. M.; Şengül, S., Modeling with fractional difference equations, J. Math. Anal. Appl., 369, 1-9 (2010) · Zbl 1204.39004
[14] Atici, F. M.; Eloe, P. W., Gronwall’s inequality in discrete fractional calculus, Comput. Math. Appl., 64, 3193-3200 (2012) · Zbl 1268.26029
[15] Dahal, R.; Goodrich, C. S., A monotonicity result for discrete fractional difference operators, Arch. Math. (Basel), 102, 293-299 (2014) · Zbl 1296.39016
[16] Ferreira, R. A.C., A discrete fractional Gronwall inequality, Proc. Amer. Math. Soc., 140, 1605-1612 (2012) · Zbl 1243.26012
[17] Goodrich, C. S., Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl., 59, 3489-3499 (2010) · Zbl 1197.39002
[18] Holm, M., Sum and difference compositions in discrete fractional calculus, Cubo, 13, 153-184 (2011) · Zbl 1248.39003
[19] Atici, F. M.; Acar, N., Exponential functions of discrete fractional calculus, Appl. Anal. Discrete Math., 7, 343-353 (2013) · Zbl 1299.39001
[20] Bastos, N. R.O.; Ferreira, R. A.C.; Torres, D. F.M., Discrete-time fractional variational problems, Signal Process., 91, 513-524 (2011) · Zbl 1203.94022
[21] Ferreira, R. A.C.; Torres, D. F.M., Fractional \(h\)-difference equations arising from the calculus of variations, Appl. Anal. Discrete Math., 5, 110-121 (2011) · Zbl 1289.39007
[22] Holm, M., Solutions to a discrete, nonlinear, \((N - 1, 1)\) fractional boundary value problem, Int. J. Dyn. Syst. Differ. Equ., 3, 267-287 (2011) · Zbl 1215.39006
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.