Advertisement

Positivity

, Volume 22, Issue 2, pp 551–573 | Cite as

Monotonicity and non-monotonicity results for sequential fractional delta differences of mixed order

  • Christopher S. Goodrich
Article
First Online:
Received:
Accepted:

Abstract

We consider the discrete fractional sequential difference \(\Delta _{1+a-\mu }^{\nu }\Delta _a^{\mu }f(t)\), where \(t\in \mathbb {N}_{3-\mu -\nu +a}\), in two separate cases, where in each case we require that \(\mu +\nu \in (1,2)\). In the first case, we show that when \(\mu \in (0,1)\) and \(\nu \in (1,2)\) it follows that the condition \(\Delta _{1+a-\mu }^{\nu }\Delta _a^{\mu }f(t)\ge 0\) implies that f is an increasing map when we impose that \(f(a)\ge 0\), \(\Delta f(a)\ge 0\), and \(\Delta f(a+1)\ge 0\). On the other hand, when \(\mu \in (1,2)\) and \(\nu \in (0,1)\) we demonstrate that the situation is very different and that this type of monotonicity result only holds when restricted to a proper subregion of the \((\mu ,\nu )\)-parameter space coupled with some additional auxiliary conditions.

Keywords

Discrete fractional calculus Monotonicity Sequential fractional delta difference 

Mathematics Subject Classification

Primary 26A48 39A70 39B62 Secondary 26A33 39A12 39A99 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Abdeljawad, T.: Dual identities in fractional difference calculus within Riemann. Adv. Differ. Equ. 19, 10 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abdeljawad, T.: On delta and nabla Caputo fractional differences and dual identities. Discrete Dyn. Nat. Soc. Art. ID 406910, 12 (2013)Google Scholar
  3. 3.
    Anastassiou, G.A.: Nabla discrete fractional calculus and nabla inequalities. Math. Comput. Model. 51, 562–571 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Atici, F.M., Acar, N.: Exponential functions of discrete fractional calculus. Appl. Anal. Discrete Math. 7, 343–353 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Atici, F.M., Eloe, P.W.: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2, 165–176 (2007)MathSciNetGoogle Scholar
  6. 6.
    Atici, F.M., Eloe, P.W.: Discrete fractional calculus with the Nabla operator. Electron. J. Qual. Theory Differ. Equ., Special Edition I, p. 12 (2009)Google Scholar
  7. 7.
    Atici, F.M., Eloe, P.W.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137, 981–989 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Atici, F.M., Eloe, P.W.: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 17, 445–456 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Atici, F.M., Eloe, P.W.: Linear systems of fractional nabla difference equations. Rocky Mt. J. Math. 41, 353–370 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Atici, F.M., Şengül, S.: Modeling with fractional difference equations. J. Math. Anal. Appl. 369, 1–9 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Atici, F.M., Uyanik, M.: Analysis of discrete fractional operators. Appl. Anal. Discrete Math. 9, 139–149 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Baoguo, J., Erbe, L., Goodrich, C.S., Peterson, A.: On the relation between delta and nabla fractional difference. Filomat 31, 1741–1753 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Erbe, L., Goodrich, C.S., Jia, B., Peterson, A.: Monotonicity results for delta fractional differences revisited. Math. Slovaca 67, 895–906 (2017)Google Scholar
  14. 14.
    Bastos, N.R.O., Mozyrska, D., Torres, D.F.M.: Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform. Int. J. Math. Comput. 11, 1–9 (2011)MathSciNetGoogle Scholar
  15. 15.
    Dahal, R., Duncan, D., Goodrich, C.S.: Systems of semipositone discrete fractional boundary value problems. J. Differ. Equ. Appl. 20, 473–491 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dahal, R., Goodrich, C.S.: A monotonicity result for discrete fractional difference operators. Arch. Math. (Basel) 102, 293–299 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dahal, R., Goodrich, C.S.: Erratum to “R. Dahal, C. S. Goodrich, A monotonicity result for discrete fractional difference operators. Arch. Math. (Basel) 102 (2014), 293–299”. Arch. Math. (Basel) 104, 599–600 (2015)Google Scholar
  18. 18.
    Dahal, R., Goodrich, C.S.: An almost sharp monotonicity result for discrete sequential fractional delta differences. J. Differ. Equ. Appl. 23, 1190–1203 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dahal, R., Drvol, S., Goodrich, C.S.: New monotonicity conditions in discrete fractional calculus with applications to extremality conditions. Analysis (Berlin) 37, 145–156 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Erbe, L., Goodrich, C.S., Jia, B., Peterson, A.: Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions. Adv. Differ. Equ. 2016, 43 (2016). doi: 10.1186/s13662-016-0760-3 MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ferreira, R.A.C.: Nontrivial solutions for fractional \(q\)-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. (70), 1–10 (2010)Google Scholar
  22. 22.
    Ferreira, R.A.C.: Positive solutions for a class of boundary value problems with fractional \(q\)-differences. Comput. Math. Appl. 61, 367–373 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ferreira, R.A.C.: A discrete fractional Gronwall inequality. Proc. Am. Math. Soc. 140, 1605–1612 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ferreira, R.A.C.: Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. J. Differ. Equ. Appl. 19, 712–718 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Goodrich, C.S.: Solutions to a discrete right-focal boundary value problem. Int. J. Differ. Equ. 5, 195–216 (2010)MathSciNetGoogle Scholar
  27. 27.
    Goodrich, C.S.: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 385, 111–124 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Goodrich, C.S.: On a first-order semipositone discrete fractional boundary value problem. Arch. Math. (Basel) 99, 509–518 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Goodrich, C.S.: A convexity result for fractional differences. Appl. Math. Lett. 35, 58–62 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Goodrich, C.S.: Systems of discrete fractional boundary value problems with nonlinearities satisfying no growth conditions. J. Differ. Equ. Appl. 21, 437–453 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Goodrich, C.S.: A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference. Math. Inequal. Appl. 19, 769–779 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Goodrich, C.S.: The relationship between discrete sequential fractional delta differences and convexity. Appl. Anal. Discrete Math. 10, 345–365 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Goodrich, C.S.: A sharp convexity result for sequential fractional delta differences. J. Differ. Equ. Appl. (2017). doi: 10.1080/10236198.2017.1380635
  34. 34.
    Goodrich, C.S.: A uniformly sharp monotonicity result for discrete fractional sequential differences. Arch. Math. (Basel). (2017). doi: 10.1007/s00013-017-1106-4
  35. 35.
    Goodrich, C., Peterson, A.C.: Discrete Fractional Calculus. Springer International Publishing, Berlin (2015). doi: 10.1007/978-3-319-25562-0 CrossRefzbMATHGoogle Scholar
  36. 36.
    Holm, M.: Sum and difference compositions and applications in discrete fractional calculus. Cubo 13, 153–184 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Jia, B., Erbe, L., Peterson, A.: Two monotonicity results for nabla and delta fractional differences. Arch. Math. (Basel) 104, 589–597 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Jia, B., Erbe, L., Peterson, A.: Convexity for nabla and delta fractional differences. J. Differ. Equ. Appl. 21, 360–373 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Jia, B., Erbe, L., Jia, B., Peterson, A.: Some relations between the Caputo fractional difference operators and integer order differences. Electron. J. Differ. Equ. No. 163, pp. 1–7 (2015)Google Scholar
  40. 40.
    Jia, B., Erbe, L., Peterson, A.: Asymptotic behavior of solutions of fractional nabla \(q\)-difference equations (submitted)Google Scholar
  41. 41.
    Jia, B., Erbe, L., Peterson, A.: Comparison theorems and asymptotic behavior of solutions of discrete fractional equations. Electron. J. Qual. Theory Differ. Equ., Paper No. 89, p. 18 (2015)Google Scholar
  42. 42.
    Jia, B., Erbe, L., Peterson, A.: Comparison theorems and asymptotic behavior of solutions of Caputo fractional equations. Int. J. Differ. Equ. 11, 163–178 (2016)MathSciNetGoogle Scholar
  43. 43.
    Lizama, C., Murillo-Arcila, M.: \(\ell _p\)-maximal regularity for a class of fractional difference equations on UMD spaces: the case \(1<\alpha \le 2\). Banach J. Math. Anal. 11, 188–206 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Lv, Z., Gong, Y., Chen, Y.: Multiplicity and uniqueness for a class of discrete fractional boundary value problems. Appl. Math. 59, 673–695 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sitthiwirattham, T., Tariboon, J., Ntouyas, S.K.: Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions. Adv. Differ. Equ. No. 296, p. 13 (2013)Google Scholar
  46. 46.
    Sitthiwirattham, T.: Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three-point fractional sum boundary conditions. Math. Methods Appl. Sci. 38, 2809–2815 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Wu, G., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Xu, R., Zhang, Y.: Generalized Gronwall fractional summation inequalities and their applications. J. Inequal. Appl. No. 242, p. 10 (2015)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsCreighton Preparatory SchoolOmahaUSA

Personalised recommendations

Cite article

Buy options