Skip to main content
SpringerLink
Log in
Book cover

Advances in Modelling and Control of Non-integer-Order Systems pp 15–24Cite as

Viable Solutions to Fractional Difference and Differential Equations

  • Conference paper

  • 1218 Accesses

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 320))

Abstract

The authors’ purpose is to consider and formulate conditions providing the existence of viable solutions to a discrete fractional equation via viability properties of fractional differential equations. We show that the existence of viable solutions to a fractional differential equation suffices to get viable solutions to a difference fractional equation.

This is a preview of subscription content, 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   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Almeida, R., Torres, D.F.M.: Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1490–1500 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aubin, J.P.: Viability Theory. Birkhäuser, Berlin (1991)

    MATH  Google Scholar 

  3. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser Boston Inc., Boston (1990)

    MATH  Google Scholar 

  4. Bai, Z., Lü, H.: Positive solutions for boundary value problems of nonlinear fractional differential equations. Journal of Mathematical Analysis and Applications 311(2), 495–505 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 29(2), 417–437 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Benchora, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order. Surveys in Mathematics and its Applications 3, 1843–7265 (2008) (print)

    Google Scholar 

  7. Cañada, A., Drábek, P., Fonda, A.: Handbook of Differential Equations Ordinary Differential Equations. Elsevier, North Holland (2005)

    MATH  Google Scholar 

  8. Ferreira, R.A.C., Torres, D.F.M.: Fractional h-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5(1), 110–121 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Girejko, E., Mozyrska, D., Wyrwas, M.: A sufficient condition of viability for fractional differential equations with the Caputo derivative. Journal of Mathematical Analysis and Applications 381 (2011)

    Google Scholar 

  10. Holm, M.T.: The theory of discrete fractional calculus: Development and application. PhD thesis, University of Nebraska - Lincoln (2011)

    Google Scholar 

  11. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North–Holland Mathematics Studies, vol. 204. Elsevier Science B. V., Amsterdam (2006)

    Book  MATH  Google Scholar 

  12. Lakshmikantham, V., Vatsala, A.S.: General uniqueness and monotone iterative technique for fractional differential equations. Applied Mathematics Letters 21, 828–834 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. 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. Nihon University, Kōriyama (1988)

    Google Scholar 

  14. Mozyrska, D., Girejko, E.: Overview of the fractional h-difference operators. In: Advances in Harmonic Analysis and Operator Theory: The Stefan Samko Anniversary Volume, vol. 229, pp. 253–267. Springer (2013)

    Google Scholar 

  15. Mozyrska, D., Girejko, E., Wyrwas, M.: A necessary condition of viability for fractional differential equations with initialization. Computers and Mathematics with Applications 62(9), 3642–3647 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mozyrska, D., Girejko, E., Wyrwas, M.: Fractional nonlinear systems with sequential operators. Central European Journal of Physics, 9 (2013), doi:10.2478/s11534-013-0223-3

    Google Scholar 

  17. Podlubny, I.: Fractional Differential Equations. Mathematics in Sciences and Engineering, vol. 198. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  18. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers S.A., Yverdon (1993)

    MATH  Google Scholar 

  19. Zhang, S.: Existence of a positive solution for some class of nonlinear fractional differential equations. Journal of Mathematical Analysis and Applications 278, 136–148 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bialystok University of Technology, Wiejska 45A, 15-351, Białystok, Poland

    Ewa Girejko, Dorota Mozyrska & Małgorzata Wyrwas

Authors
  1. Ewa Girejko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dorota Mozyrska
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Małgorzata Wyrwas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ewa Girejko .

Editor information

Editors and Affiliations

  1. Department of Electrical, Control and Computer Engineering, Opole University of Technology, Opole, Poland

    Krzysztof J. Latawiec

  2. Department of Electrical, Control and Computer Engineering, Opole University of Technology, Opole, Poland

    Marian Łukaniszyn

  3. Department of Electrical, Control and Computer Engineering, Opole University of Technology, Opole, Poland

    Rafał Stanisławski

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Girejko, E., Mozyrska, D., Wyrwas, M. (2015). Viable Solutions to Fractional Difference and Differential Equations. In: Latawiec, K., Łukaniszyn, M., Stanisławski, R. (eds) Advances in Modelling and Control of Non-integer-Order Systems. Lecture Notes in Electrical Engineering, vol 320. Springer, Cham. https://doi.org/10.1007/978-3-319-09900-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-09900-2_2

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-09899-9

  • Online ISBN: 978-3-319-09900-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation