Skip to main content
SpringerLink
Log in

Study of existence results for fractional functional differential equations involving Riesz-Caputo derivative

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

Within this work, we look into the existence results for a family of fractional functional differential equations employing the Riesz-Caputo fractional derivative in a Banach space. Fractional calculus techniques, Kuratowski’s measure of non-compactness, Carath\(\acute{e}\)odory conditions, and some theorems on fixed points are used to establish existence results. In the end, a few examples are showcased to evince the proficiency of the offered results.

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

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no data sets were generated or analysed duringthe current study.

References

  1. Kilbas, A., H. Srivastava, and J. Trujillo. 2006. Theory and applications of fractional differential equations. Amsterdam: Elsevier.

    Google Scholar 

  2. Podlubny, I. 1998. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198. San Diego: Elsevier.

    Google Scholar 

  3. Kumar, S., D. Kumar, S. Abbasbandy, and M. Rashidi. 2014. Analytical solution of fractional Navier-Stokes equation by using modified Laplace decomposition method. Ain Shams Engineering Journal 5 (2): 569–574.

    Article  Google Scholar 

  4. Kumar, S., R.K. Pandey, K. Kumar, S. Kamal, and T.N. Dinh. 2022. Finite difference-collocation method for the generalized fractional diffusion equation. Fractal and Fractional 6 (7): 387.

    Article  Google Scholar 

  5. Yadav, S., R.K. Pandey, and P.K. Pandey. 2021. Numerical approximation of tempered fractional Sturm-Liouville problem with application in fractional diffusion equation. International Journal for Numerical Methods in Fluids 93 (3): 610–627.

    Article  MathSciNet  Google Scholar 

  6. Changpin, L., and Z. Fanhai. 2015. Numerical methods for fractional calculus, vol. 24. New York: CRC Press.

    Google Scholar 

  7. Shen, J., T. Tang, and L.-L. Wang. 2011. Spectral methods: algorithms, analysis and applications, vol. 41. New York: Springer.

    Google Scholar 

  8. Alqhtani, M., K.M. Owolabi, K.M. Saad, and E. Pindza. 2022. Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology. Chaos, Solitons & Fractals 161: 112394.

    Article  MathSciNet  Google Scholar 

  9. Shukla, A.K., R.K. Pandey, and S. Yadav. 2021. Adaptive fractional masks and super resolution based approach for image enhancement. Multimedia Tools and Applications 80 (20): 30213–30236.

    Article  Google Scholar 

  10. Agrawal, O.P. 2012. Some generalized fractional calculus operators and their applications in integral equations. Fractional Calculus and Applied Analysis 15 (4): 700–711.

    Article  MathSciNet  Google Scholar 

  11. Jumarie, G. 2006. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications 51 (9–10): 1367–1376.

    Article  MathSciNet  Google Scholar 

  12. Mayergoyz, I.D. 2003. Mathematical models of hysteresis and their applications. New York: Academic Press.

    Google Scholar 

  13. Arino, O., M.L. Hbid, and E.A. Dads. 2007. Delay Differential Equations and Applications. In Proceedings of the NATO Advanced Study Institute Held in Marrakech, Morocco, 9–21 September 2002 vol. 205. Springer, Morocco

  14. Kuang, Y. 1993. Delay differential equations: with applications in population dynamics. San Diego: Academic Press.

    Google Scholar 

  15. Smith, H.L. 2011. An introduction to delay differential equations with applications to the life sciences, vol. 57. New York: Springer.

    Google Scholar 

  16. Rihan, F.A. 2021. Delay differential equations and applications to biology. Singapore: Springer.

    Book  Google Scholar 

  17. Zhu, B., L. Liu, and Y. Wu. 2016. Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay. Applied Mathematics Letters 61: 73–79.

    Article  MathSciNet  Google Scholar 

  18. Zhu, B., L. Liu, and Y. Wu. 2019. Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay. Computers & Mathematics with Applications 78 (6): 1811–1818.

    Article  MathSciNet  Google Scholar 

  19. Zhu, B., B.-Y. Han, and W.-G. Yu. 2020. Existence of mild solutions for a class of fractional non-autonomous evolution equations with delay. Acta Mathematicae Applicatae Sinica, English Series 36 (4): 870–878.

    Article  MathSciNet  Google Scholar 

  20. Zhu, B., B.-Y. Han, and W.-g Yu. 2020. Existence of mild solutions for a class of fractional non-autonomous evolution equations with delay. Acta Mathematicae Applicatae Sinica, English Series 36 (4): 870–878.

    Article  MathSciNet  Google Scholar 

  21. Mishra, K.K., S. Dubey, and D. Baleanu. 2022. Existence and controllability of a class of non-autonomous nonlinear evolution fractional integrodifferential equations with delay. Qualitative Theory of Dynamical Systems 21 (4): 1–22.

    Article  MathSciNet  Google Scholar 

  22. Zaslavsky, G.M. 2002. Chaos, fractional kinetics, and anomalous transport. Physics Reports 371 (6): 461–580.

    Article  MathSciNet  Google Scholar 

  23. Metzler, R., and J. Klafter. 2000. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 339 (1): 1–77.

    Article  MathSciNet  CAS  Google Scholar 

  24. Pandey, R.K., O.P. Singh, and V.K. Baranwal. 2011. An analytic algorithm for the space-time fractional advection-dispersion equation. Computer Physics Communications 182 (5): 1134–1144.

    Article  MathSciNet  CAS  Google Scholar 

  25. Ye, H., F. Liu, V. Anh, and I. Turner. 2014. Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations. Applied Mathematics and Computation 227: 531–540.

    Article  MathSciNet  Google Scholar 

  26. Luchko, Y. 2011. Maximum principle and its application for the time-fractional diffusion equations. Fractional Calculus and Applied Analysis 14 (1): 110–124.

    Article  MathSciNet  Google Scholar 

  27. Bo Zhu, B.H. 2022. Approximate controllability for mixed type non-autonomous fractional differential equations. Qualitative Theory of Dynamical Systems 21 (4): 111.

    Article  MathSciNet  Google Scholar 

  28. Zhu, B., B. Han, L. Liu, and W. Yu. 2020. On the fractional partial integro-differential equations of mixed type with non-instantaneous impulses. Boundary Value Problems 2020: 154.

    Article  MathSciNet  Google Scholar 

  29. Chen, F., D. Baleanu, and G.-C. Wu. 2017. Existence results of fractional differential equations with Riesz-Caputo derivative. The European Physical Journal Special Topics 226 (16): 3411–3425.

    Article  Google Scholar 

  30. Gu, C.-Y., J. Zhang, and G.-C. Wu. 2019. Positive solutions of fractional differential equations with the Riesz space derivative. Applied Mathematics Letters 95: 59–64.

    Article  MathSciNet  Google Scholar 

  31. Chen, F., A. Chen, and X. Wu. 2019. Anti-periodic boundary value problems with Riesz-Caputo derivative. Advances in Difference Equations 2019 (1): 1–13.

    Article  MathSciNet  Google Scholar 

  32. Naas, A., M. Benbachir, M.S. Abdo, and A. Boutiara. 2022. Analysis of a fractional boundary value problem involving Riesz-Caputo fractional derivative. Advances in the Theory of Nonlinear Analysis and its Application 1 (1): 14–27.

    Google Scholar 

  33. Chen Pengyu, L.Y., and Zhang Xuping. 2018. A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Communications on Pure and Applied Analysis 17 (5): 1975–1992.

    Article  MathSciNet  Google Scholar 

  34. Chen Pengyu, L.Y., and Z. Xuping. 2020. Cauchy problem for fractional non-autonomous evolution equations. Banach Journal of Mathematical Analysis 14 (2): 559–584.

    Article  MathSciNet  Google Scholar 

  35. Chen Pengyu, L.Y., and Z. Xuping. 2020. Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators. Fractional Calculus and Applied Analysis 23 (1): 268–291.

    Article  MathSciNet  Google Scholar 

  36. Chen Pengyu, Z.X., and L. Yongxiang. 2021. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete and Continuous Dynamical Systems - Series B 26 (3): 1531–1547.

    Article  MathSciNet  Google Scholar 

  37. Granas, A., and J. Dugundji. 2003. Fixed point theory, vol. 14. Vancouver: Springer.

    Book  Google Scholar 

  38. Fan, Z., and G. Li. 2010. Existence results for semilinear differential equations with nonlocal and impulsive conditions. Journal of Functional Analysis 258 (5): 1709–1727.

    Article  MathSciNet  Google Scholar 

  39. Akhmerov, R.R., M.I. Kamenskii, A.S. Potapov, A. Rodkina, and B.N. Sadovskii. 1992. Measures of noncompactness and condensing operators, vol. 55. Vancouver: Springer.

    Book  Google Scholar 

  40. Banaś, J. 1980. On measures of noncompactness in banach spaces. Commentationes Mathematicae Universitatis Carolinae 21 (1): 131–143.

    MathSciNet  Google Scholar 

  41. Deimling, K. 2010. Nonlinear functional analysis. Berlin: Courier Corporation.

    Google Scholar 

  42. Liu, L., F. Guo, C. Wu, and Y. Wu. 2005. Existence theorems of global solutions for nonlinear Volterra type integral equations in banach spaces. Journal of Mathematical Analysis and Applications 309 (2): 638–649.

    Article  MathSciNet  Google Scholar 

  43. Chen, P., and Y. Li. 2013. Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions. Results in Mathematics 63 (3): 731–744.

    Article  MathSciNet  Google Scholar 

  44. Lishan, L., et al. 1996. Iterative method for solutions and coupled quasi-solutions of nonlinear Fredholm integral equations in ordered banach spaces. Indian Journal of Pure and Applied Mathematics 27: 959–972.

    MathSciNet  Google Scholar 

  45. Liu, L. 2000. Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in banach spaces. Nonlinear Analysis: Theory, Methods & Applications 42 (4): 583–598.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors express their gratitude to the reviewers for their valuable comments incorporated in the revised version of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varansi, Uttar Pradesh, 221005, India

    Pratima Tiwari & Rajesh K. Pandey

  2. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

    D. N. Pandey

Authors
  1. Pratima Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajesh K. Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. N. Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rajesh K. Pandey.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Communicated by S Ponnusamy.

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 (e.g. a society or other partner) 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

Tiwari, P., Pandey, R.K. & Pandey, D.N. Study of existence results for fractional functional differential equations involving Riesz-Caputo derivative. J Anal (2024). https://doi.org/10.1007/s41478-024-00728-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s41478-024-00728-1

Keywords

Mathematics Subject Classification

Search

Navigation