Skip to main content
SpringerLink
Log in

Qualitative Investigation of Nonlinear Fractional Coupled Pantograph Impulsive Differential Equations

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this manuscript a qualitative analysis to a nonlinear coupled system of pantograph impulsive fractional differential equations (PIFDEs) is established. By the use of Banach and Krasnoselskii’s fixed-point theorems some adequate conditions for the existence and uniqueness of solution to the considered problem are developed. The advantage of using Krasnoselskii’s fixed-point theorem is that it uses slight relax compact conditions as compared to other fixed point results. Furthermore, the manuscript is enriched by adding some results about Ulam–Hyers type stability. Finally, with the help of pertinent examples, the obtained theoretical results are justified.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data Availability

The Data has used in this article is included within the article.

References

  1. Ercan, A., Kavvas, M.L.: Numerical evaluation of fractional vertical soil water flow equations. Water 2021(13), 511 (2021)

    Article  Google Scholar 

  2. Moshrefi-Torbati, M., Hammond, J.K.: Physical and geometrical interpretation of fractional operators. J. Frankl. Inst. 335(6), 1077–1086 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Tarasov, V.E.: Geometric interpretation of fractional-order derivative. Fract. Cal. Appl. Anal. 19(5), 1200–1221 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. ArXiv:math/0110241 (2001)

  5. Tarasov, V.E.: Interpretation of fractional derivatives as reconstruction from sequence of integer derivatives. Fundamenta Informaticae 151(1–4), 431–442 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hilfer, R.: Mathematical and Physical Interpretations of Fractional Derivatives and Integrals. Handbook of Fractional Calculus with Applications 1, 47–85 (2019)

  7. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  8. Ortigueira, M.D., Machado, J.T.: What is a fractional derivative? J. Comput. Phys. 293, 4–13 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dalir, M., Bashour, M.: Applications of fractional calculus. Appl. Math. Sci. 4(21), 1021–1032 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Benchohra, M., Graef, J.R., Hamani, S.: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 87, 851–863 (2010)

    Article  MATH  Google Scholar 

  11. Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative. Adv. Differ. Equ. 2009, 1–47 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Wang, G., Liu, S., Baleanu, D., Zhang, L.: A new impulsive multi-orders fractional differential equation involving multipoint fractional integral boundary conditions. Abstr. Appl. Anal. 2014, 932747, 10 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Balachandran, K., Kiruthika, S., Trujillo, J.: Existence of solutions of nonlinear fractional pantograph equations. Acta Mathematica Scientia 33(3), 712–720 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yukunthorn, W., Ntouyas, S.K., Tariboon, J.: Impulsive multiorders Riemann–Liouville fractional differential equations. Discret. Dyn. Nat. Soc. Article ID 603893, 9 (2015)

  15. Benchohra, M., Bouriah, S., Nieto, J.J.: Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative. Demonstr. Math. 52(1), 437–450 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lakshmikanthan, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  Google Scholar 

  17. Chernousko, F., Akulenko, L., Sokolov, B.: Control of Oscillations. Nauka, Moskow (1980)

    Google Scholar 

  18. Bainov, D., Dimitrova, M., Dishliev, A.: Oscillation of the bounded solutions of impulsive differential-difference equations of second order. Appl. Math. Comput. 114(1), 61–68 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chua, L., Yang, L.: Cellular neural networks: applications. IEEE Trans. Circuits Syst. CAS 35, 1273–1290 (1988)

    Article  MathSciNet  Google Scholar 

  20. Babitskii, V., Krupenin, V.: Vibration in Strongly Nonlinear Systems. Nauka, Moskow (1985)

    Google Scholar 

  21. Andronov, A., Witt, A., Haykin, S.: Oscilation Theory. Nauka, Moskow (1981)

    Google Scholar 

  22. Popov, E.: The Dynamics of Automatic Control Systems. Gostehizdat, Moskow (1964)

    Google Scholar 

  23. Zavalishchin, S., Sesekin, A.: Impulsive Processes: Models and Applications. Nauka, Moskow (1991)

    MATH  Google Scholar 

  24. Hale, J.K., Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (2013)

    MATH  Google Scholar 

  25. Ockendon, J.R., Taylor, A.B.: The dynamics of a current collection system for an electric locomotive. Proc. R Soc. Lond. Ser. A 322, 447–468 (1971)

    Article  Google Scholar 

  26. Rossetti, M., Bardella, P., Montrosset, I.: Modeling passive mode-locking in quantum dot lasers: a comparison between a finite-difference traveling-wave model and a delayed differential equation approach. IEEE J. Quant. Electron. 47(5), 569–576 (2011)

    Article  Google Scholar 

  27. Hofer, P., Lion, A.: Modelling of frequency- and amplitude-dependent material properties of filler-reinforced rubber. J. Mech. Phys. Solids 57, 500–520 (2009)

    Article  MATH  Google Scholar 

  28. Sedaghat, S., Ordokhani, Y., Dehghan, M.: Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Commun. Nonl. Sci. Numer. Simul. 17(12), 4815–4830 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Iqbal, M., Shah, K., Khan, R.A.: On using coupled fixed point theorems for mild solutions to coupled system of multi-point boundary value problems of nonlinear fractional hybrid pantograph differential equations. Math. Meth. Appl. Sci. 44, 1–14 (2019)

    Google Scholar 

  30. Griebel, T.: The pantograph equation in quantum calculus. Missouri University of Science and Technology (2017)

  31. Hyers, D.H.: On the stability of the linear functional equations. Proc. Natl. Acad. Sci. USA 27(4), 222–224 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ulam, S.M.: A Collection of the Mathematical Problems. Interscience Publishers, New York (1960)

    MATH  Google Scholar 

  33. Jung, S.M.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17(10), 1135–1140 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zada, A., Ali, W.: Choonkil, Ulam type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type. Appl. Math. Comput. 350, 60–65 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, J., Lv, L., Zhou, W.: Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 63, 1–10 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Kate, T., McLeod, J.B.: The functional-differential equation \( y^{\prime }(x)= ay({\lambda x})+ by(x)\). Bull. Am. Math. Soc. 77(6), 891–937 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  37. Urs, C.: Coupled fixed point theorem and applications to periodic boundary value problem. Miskolc. Math. Notes 14(1), 323–333 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ray, S.S., Atangana, A., Noutchie, S.C., Kurulay, M., Bildik, N., Kilicman, A.: Fractional calculus and its applications in applied mathematics and other sciences. Mathematical Problems in Engineering, Article ID 849395, 2 (2014). https://doi.org/10.1155/2014/849395

  39. Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii’s type. Math. Nachr. 189, 23–31 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  40. Bainov, D.D., Simenonv, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow (1993)

    Google Scholar 

  41. Jackson, D.: Existence and uniqueness of solutions of a semilinear nonlocal parabolic equations. J. Math. Anal. Appl. 172, 256–265 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, L., Chen, H.: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. Article ID 404917, 14 (2011)

  43. Guo, T., Wei, J.: Impulsive problems for fractional differential equations with boundary value conditions. Comput. Math. Appl. 64(10), 3281–3291 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors Kamal Shah and Thabet Abdeljawad would like to thank Prince Sultan University for support through TAS research lab.

Author information

Authors and Affiliations

  1. Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia

    Kamal Shah & Thabet Abdeljawad

  2. Department of Mathematics, University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhwa, 18000, Pakistan

    Kamal Shah

  3. Department of Mathematics, Government Post Graduate Jahanzeb College, Saidu Sharif, Swat, Khyber Pakhtunkhwa, Pakistan

    Israr Ahmad

  4. Departamento de Estatistica, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

    Juan J. Nieto

  5. Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan

    Ghaus Ur Rahman

  6. Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

    Thabet Abdeljawad

Authors
  1. Kamal Shah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Israr Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Juan J. Nieto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ghaus Ur Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Thabet Abdeljawad
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

K.S. designed the problem and use methodology. I.A. performed formal analysis, J.J.N. edited the draft initially. G.R. updated the literature review. T.A. has contributed in the revised version by updating literature and answering the comments of referees together with other authors.

Corresponding authors

Correspondence to Kamal Shah or Thabet Abdeljawad.

Ethics declarations

Conflict of Interest

There does not exist conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shah, K., Ahmad, I., Nieto, J.J. et al. Qualitative Investigation of Nonlinear Fractional Coupled Pantograph Impulsive Differential Equations. Qual. Theory Dyn. Syst. 21, 131 (2022). https://doi.org/10.1007/s12346-022-00665-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00665-z

Keywords

Mathematics Subject Classification

Search

Navigation