Skip to main content
SpringerLink
Log in

Caputo Nabla Fractional Boundary Value Problems

  • Conference paper
  • First Online:
Progress on Difference Equations and Discrete Dynamical Systems (ICDEA 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 341))

Included in the following conference series:

  • 341 Accesses

Abstract

We study boundary value problems with the Caputo nabla difference in the context of discrete fractional nabla calculus, especially when the right boundary condition has a fractional order. We first construct the Green’s function for the general case and study the properties of the Green’s function in several cases. We then apply the cone theory in a Banach space to show the existence of positive solutions to a nonlinear boundary value problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ahrendt, K., Castle, L., Holm, M., Yochman, K.: Laplace transforms for the nabla-difference operator and a fractional variation of parameters formula. Commun. Appl. Anal. 16(3), 317 (2012)

    Google Scholar 

  2. Ahrendt, K.: The Existence of Solutions for a Nonlinear, Fractional Self-Adjoint Difference Equation. Ph.D. thesis, University of Nebraska-Lincoln, 2017

    Google Scholar 

  3. Atıcı, F.M., Eloe, P.W.: Two-point boundary value problems for finite fractional difference equations. J. Diff. Equ. Appl. 17(04), 445–456 (2011)

    Article  MathSciNet  Google Scholar 

  4. Atıcı, F.M., Uyanik, M.: Analysis of discrete fractional operators. Appl. Anal. Discret. Mat., 139–149 (2015)

    Google Scholar 

  5. Atıcı, F.M., Belcher, M., Marshall, D.: A new approach for modeling with discrete fractional equations. Fundam. Inf. 151(1–4), 313–324 (2017)

    Article  MathSciNet  Google Scholar 

  6. Brackins, A.M.: Boundary Value Problems of Nabla Fractional Difference Equations. Ph.D. thesis, University of Nebraska-Lincoln (2015)

    Google Scholar 

  7. Eloe, P.W., Kublik C.M., Neugebauer, J.T.: Comparison of green’s functions for a family of boundary value problems for fractional difference equations. J. Diff. Equ. Appl., 1–12 (2018)

    Google Scholar 

  8. Erbe, L., Peterson, A.: Positive solutions for a nonlinear differential equation on a measure chain. Math. Comput. Model. 32(5–6), 571–585 (2000)

    Article  MathSciNet  Google Scholar 

  9. 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. Diff. Equ. 2016(1), 43 (2016)

    Google Scholar 

  10. Gensler, S.C.: Fractional Difference Operators and Relatedlems. Ph.D. thesis, University of Nebraska-Lincoln (2018)

    Google Scholar 

  11. Goodrich, C.S.: On a fractional boundary value problem with fractional boundary conditions. Appl. Math. Lett. 25(8), 1101–1105 (2012)

    Article  MathSciNet  Google Scholar 

  12. Goodrich, C., Peterson, A.C.: Discrete Fractional Calculus. Springer (2015)

    Google Scholar 

  13. 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)

    MathSciNet  MATH  Google Scholar 

  14. Graef, J.R., Ho, S., Kong, L., Wang, M.: A fractional differential equation model for bike share systems. The Joint Mathematics Meetings, Baltimore, MD, January 2019

    Google Scholar 

  15. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press (1988)

    Google Scholar 

  16. Holm, M.T.: The theory of Discrete Fractional Calculus: Development and Application. Ph.D. thesis, University of Nebraska-Lincoln (2011)

    Google Scholar 

  17. Ikram, A.: Green’s Functions and Lyapunov Inequalities for Nabla Caputo Boundary Value Problems. Ph.D. thesis, University of Nebraska-Lincoln (2018)

    Google Scholar 

  18. Jia, B., Erbe, L., Goodrich, C., Peterson, A.: The relation between nabla fractional differences and nabla integer differences. Filomat 31(6), 1741–1753 (2017)

    Article  MathSciNet  Google Scholar 

  19. Krasnosel’skiĭ, M.A.: Positive solutions of operator equations. P. Noordhoff (1964)

    Google Scholar 

  20. Machado, J.T., Silva, M.F., Barbosa, R.S., Jesus, I.S., Reis, C.M., Marcos, M.G., Galhano, A.F.: Some applications of fractional calculus in engineering. Math. Probl. Eng. (2010)

    Google Scholar 

  21. Miller, K.S., Ross, B.: Fractional difference calculus. In: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139–152 (1988)

    Google Scholar 

  22. Setniker, A.: Sequential Differences in Nabla Fractional Calculus. Ph.D. thesis, University of Nebraska-Lincoln (2019)

    Google Scholar 

  23. St Goar, J.: A Caputo Boundary Value Problem in Nabla Fractional Calculus. Ph.D. thesis, University of Nebraska-Lincoln (2016)

    Google Scholar 

  24. Valério, D., Machado, J.T., Kiryakova, V.: Some pioneers of the applications of fractional calculus. Fract. Calc. Appl. Anal. 17(2), 552–578 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Nebraska-Lincoln, Lincoln, NE, 68588, USA

    ALLAN PETERSON

  2. Tusculum University, Greeneville, TN, 37615, USA

    Wei Hu

Authors
  1. ALLAN PETERSON
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ALLAN PETERSON .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University College London, London, UK

    Steve Baigent

  2. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

    Martin Bohner

  3. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

PETERSON, A., Hu, W. (2020). Caputo Nabla Fractional Boundary Value Problems. In: Baigent, S., Bohner, M., Elaydi, S. (eds) Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-030-60107-2_1

Download citation

Publish with us

Policies and ethics

Search

Navigation