Skip to main content
SpringerLink
Log in

Hyers–Ulam stability of random functional differential equation involving fractional-order derivative

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we investigate the existence of solutions for the random fractional differential equations (RFDEs) with delay by employing Banach and Schaefer’s fixed point theorems. In addition, we present the Hyers–Ulam stability (HUS) and Hyers–Ulam–Rassias stability (HURS) for RFDEs with delay through using Gronwall inequality. Two examples are given to visualize theoretical 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

References

  • An TV, Phu ND, Van Hoa N (2022) A survey on non-instantaneous impulsive fuzzy differential equations involving the generalized Caputo fractional derivative in the short memory case. Fuzzy Sets Syst 443:160–197. https://www.sciencedirect.com/science/article/pii/S0165011421003870

  • Aubin JP, Ekeland I (2006) Applied nonlinear analysis. Courier Corporation, Chelmsford

    MATH  Google Scholar 

  • Burgos C, Cortés J, Villafuerte L, Villanueva R (2017) Mean square calculus and random linear fractional differential equations: theory and applications. Appl Math Nonlinear Sci 2(2):317–328. https://sciendo.com/it/article/10.21042/AMNS.2017.2.00026

  • Burgos C, Cortés J-C, Villafuerte L, Villanueva R-J (2020) Mean square convergent numerical solutions of random fractional differential equations: approximations of moments and density. J Comput Appl Math 378:112925

    Article  MathSciNet  Google Scholar 

  • Duc TM, Van Hoa N (2021) Stabilization of impulsive fractional-order dynamic systems involving the Caputo fractional derivative of variable-order via a linear feedback controller. Chaos Solitons Fractals 153:111525

    Article  MathSciNet  Google Scholar 

  • Duc TM, Van Hoa N, Dao T-P (2018) Adaptive fuzzy fractional-order nonsingular terminal sliding mode control for a class of second-order nonlinear systems. J Comput Nonlinear Dyn 13(3):031004 (8 pages). https://asmedigitalcollection.asme.org/computationalnonlinear/article-abstract/13/3/031004/380665/Adaptive-Fuzzy-Fractional-Order-Nonsingular

  • El-Sayed A, Eladdad EE, Madkour HFA (2015) On some equivalent problems of stochastic differential equations of fractional order. Fract Calc Appl 6(2):115–122

    MathSciNet  MATH  Google Scholar 

  • El-Sayed A, Gaafar F, El-Gendy M (2017) Continuous dependence of the solution of random fractional-order differential equation with nonlocal conditions. Fract Differ Calc 7(1):135–149

    Article  MathSciNet  Google Scholar 

  • Hafiz FM (2004) The fractional calculus for some stochastic processes. Stoch Anal Appl 22(2):507–523

    Article  MathSciNet  Google Scholar 

  • Hafiz FM, El-Sayed AM, El-Tawil MA (2001) On a stochastic fractional calculus. Fract Calc Appl Anal 4(1):81–90

    MathSciNet  MATH  Google Scholar 

  • Hoa NV, Duc TM, Vu H (2019) A new result on fractional differential inequality and applications to control of dynamical systems. J Dyn Syst Meas Control 141(9):94501. https://asmedigitalcollection.asme.org/dynamicsystems/article-abstract/141/9/094501/726560/A-New-Result-on-Fractional-Differential-Inequality?redirectedFrom=fulltext

  • Khudair AR, Haddad SAM, Khalaf SL (2016) Mean square solutions of second-order random differential equations by using the differential transformation method. Open J Appl Sci 6:287–297

    Article  Google Scholar 

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. North-Holland mathematics studies. Elsevier Science Inc., Amsterdam

    Book  Google Scholar 

  • Ladde G, Lakshmikantham V (1980) Random differential inequalities. Academic Press, New York

    MATH  Google Scholar 

  • Lakshmikantham V, Leela S, Devi JV (2009) Theory of fractional dynamic systems. Cambridge Scientific Publishers, Cambridge

    MATH  Google Scholar 

  • Long HV, Kim Son NT, Thanh Tam HT, Yao J-C (2017) Ulam stability for fractional partial integro-differential equation with uncertainty. Acta Math Vietnam 42(4):675–700

    Article  MathSciNet  Google Scholar 

  • Lupulescu V, Ntouyas SK (2012) Random fractional differential equations. Int Electron J Pure Appl Math 4(2):119–136

    MathSciNet  MATH  Google Scholar 

  • Lupulescu V, O’Regan D, ur Rahman G (2014) Existence results for random fractional differential equations. Opusc Math 34(4):813–825

    Article  MathSciNet  Google Scholar 

  • Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience, New York

    MATH  Google Scholar 

  • Nguyen SD, Lam BD, Ngo VH (2020) Fractional-order sliding-mode controller for semi-active vehicle MRD suspensions. Nonlinear Dyn 101(2):795–821

    Article  Google Scholar 

  • Nouri K, Ranjbar H (2015) Mean square convergence of the numerical solution of random differential equations. Mediterr J Math 12(3):1123–1140

    Article  MathSciNet  Google Scholar 

  • O’Regan D, Van Hoa N (2020) An initial value problem involving Caputo–Hadamard fractional derivative: the extremal solutions and stabilization. J Adv Eng Comput 4(2):149–161

    Article  Google Scholar 

  • Phu ND, Lupulescu V, Van Hoa N (2021) Neutral fuzzy fractional functional differential equations. Fuzzy Sets Syst 419:1–34

    Article  MathSciNet  Google Scholar 

  • Shuorui Z, Jitao S (2016) On existence and uniqueness of random impulsive differential equations. J Syst Sci Complex 29(2):300–314

    Article  MathSciNet  Google Scholar 

  • Slimane I, Dahmani Z (2021) A continuous and fractional derivative dependance of random differential equations with nonlocal conditions. J Interdiscip Math 01–14:2021

    Google Scholar 

  • Smart DR (1980) Fixed point theorems, vol 66. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Soong TT (1973) Random differential equations in science and engineering. Academic Press, New York

    MATH  Google Scholar 

  • Strand JL (1985) Random ordinary differential equations. Reidel, Boston

    MATH  Google Scholar 

  • Sun H, Chen Y, Chen W (2011) Random-order fractional differential equation models. Signal Process 91(3):525–530

    Article  MathSciNet  Google Scholar 

  • Tobias N, Florian R (2013) Random differential equations in scientific computing. De Gruyter Open, Berlin

    MATH  Google Scholar 

  • Vinh An T, Vu H, Van Hoa N (2022) Finite-time stability of fractional delay differential equations involving the generalized Caputo fractional derivative with non-instantaneous impulses. Math Methods Appl Sci 45(9):4938–4955. https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.8084

  • Vu H (2016) Random fractional functional differential equations. Int J Nonlinear Anal Appl 7(2):253–267

    MathSciNet  MATH  Google Scholar 

  • Vu H, Hoa NV (2020) On initial value problem of random fractional differential equation with impulses. Hacettepe J Math Stat 49(1):282–293. https://dergipark.org.tr/tr/download/article-file/683806

  • Vu H, Phung NN, Phuong N (2016) On fractional random differential equations with delay. Opusc Math 36(4):541–556

    Article  MathSciNet  Google Scholar 

  • Yang D, Wang J (2017) Non-instantaneous impulsive fractional-order implicit differential equations with random effects. Stoch Anal Appl 35(4):719–741

    Article  MathSciNet  Google Scholar 

  • Ye H, Gao J, Ding Y (2007) A generalized Gronwall inequality and its application to a fractional differential equation. J Math Anal Appl 328(2):1075–1081

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper.

Author information

Authors and Affiliations

  1. Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Ho Vu

  2. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Ngo Van Hoa

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Ngo Van Hoa

Authors
  1. Ho Vu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ngo Van Hoa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ngo Van Hoa.

Additional information

Communicated by Vasily E. Tarasov.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vu, H., Van Hoa, N. Hyers–Ulam stability of random functional differential equation involving fractional-order derivative. Comp. Appl. Math. 41, 204 (2022). https://doi.org/10.1007/s40314-022-01915-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01915-1

Keywords

Mathematics Subject Classification

Search

Navigation