Skip to main content

Advertisement

SpringerLink
Log in

Some New Gronwall-Type Integral Inequalities and their Applications to Finite-Time Stability of Fractional-Order Neural Networks with Hybrid Delays

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

In this paper, some Gronwall-type integral inequalities with discrete and distributed delays are explored to analyze differential or integral systems with hybrid delays. These new inequalities generalize some previous ones which have played an important part in the research on dynamic behavior of systems. Based on the established Henry-Gronwall type integral inequality, an improved criterion is derived to ensure the finite-time stability for a class of fractional-order neural networks with hybrid delays. Finally, some numerical examples are provided to show the effectiveness and the less conservativeness of the obtained criterion.

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
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Gronwall TH (1919) Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann Math 20:292–296

    MathSciNet  MATH  Google Scholar 

  2. Li ZZ, Wang WS (2019) Some nonlinear Gronwall-Bellman type retarded integral inequalities with power and their applications. Appl Math Comput 347:839–852

    MathSciNet  MATH  Google Scholar 

  3. Henry D (1981) Geometric theory of semilinear parabolic equations, lecture notes in math, vol 840. Springer-Verlag, New York/Berlin

    Google Scholar 

  4. Lipovan O (2000) A retarded Gronwall-like inequality and its applications. J Math Anal Appl 252:389–01

    MathSciNet  MATH  Google Scholar 

  5. Medve\(\breve{d}\) M (1997) A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J Math Anal Appl 214:349–66

  6. Ye HP (2007) A generalized Gronwall inequality and its application to a fractional differential equation. J Math Anal Appl 328(2):1075–1081

    MathSciNet  MATH  Google Scholar 

  7. Du FF, Lu JG (2021) New criteria for finite-time stability of fractional order memristor-based neural networks with time delays. Neurocomputing 421:349–359

    Google Scholar 

  8. Ma Q, Pecric J (2008) Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations. J Math Anal Appl 341:894–905

    MathSciNet  Google Scholar 

  9. Webb JRL (2019) Weakly singular Gronwall inequalities and applications to fractional differential equations. J Math Anal Appl 471(1–2):692–711

    MathSciNet  MATH  Google Scholar 

  10. Zhu T (2018) New Henry-Gronwall integral inequalities and their applications to fractional differential equations. Bull Braz Math Soc (NS) 49(3):647–657

    MathSciNet  MATH  Google Scholar 

  11. Wu Q (2017) A new type of the Gronwall-Bellman inequality and its application to fractional stochastic differential equations. Cogent Math 4:1279781

    MathSciNet  MATH  Google Scholar 

  12. Hussain S, Sadia H, Aslam S (2019) Some generalized Gronwall-Bellman-Bihari type integral inequalities with application to fractional stochastic differential equation. Filomat 33(3):815–824

    MathSciNet  MATH  Google Scholar 

  13. Yang XJ, Song QK, Liu YR, Zhao ZJ (2015) Finite-time stability analysis of fractional-order neural networks with delay. Neurocomputing 152:19–26

    Google Scholar 

  14. Wang LM, Song QK, Liu YR, Zhao ZJ, Alsaadi FE (2017) Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with both leakage and time-varying delays. Neurocomputing 245:86–101

    Google Scholar 

  15. Syed AM, Narayanan G, Orman Z (2020) Finite time stability analysis of fractional-order complex-valued memristive neural networks with proportional delays. Neural Process Lett 51:407–426

    Google Scholar 

  16. Syed AM, Narayanan G, Saroha S, Priya B (2021) Finite-time stability analysis of fractional-order memristive fuzzy cellular neural networks with time delay and leakage term. Math Comput Simulat 185:468–485

    MathSciNet  MATH  Google Scholar 

  17. Lazarevi\(\acute{c}\) M (2007) Finite-time stability analysis of fractional order time delay systems: Bellman-Gronwall’s approach. Sci Tech Rev 57(1):8–15

  18. Lazarevi\(\acute{c}\) M, Debeljkovi DL (2005) Finite time stability analysis of linear autonomous fractional order systems with delayed state. Asian J Control 7(4):440–447

  19. Ye HP, Gao JM (2011) Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay. Appl Math Comput 218(8):4152–4160

    MathSciNet  MATH  Google Scholar 

  20. Shao J, Meng F (2013) Gronwall-Bellman type inequalities and their applications to fractional differential equations. Abstr Appl Anal 2013:1056–1083

    MathSciNet  MATH  Google Scholar 

  21. Xu R, Meng F (2016) Some new weakly singular integral inequalities and their applications to fractional differential equations. J Inequal Appl 2016:78

    MathSciNet  MATH  Google Scholar 

  22. Yang ZY, Zhang J, Hu JH, Mei J (2021) New results on finite-time stability for fractional-order neural networks with proportional delay. Neurocomputing 442:327–336

    Google Scholar 

  23. Du FF, Lu JG (2021) New criterion for finite-time synchronization of fractional order memristor-based neural networks with time delay. Appl Math Comput 389:125616

    MathSciNet  MATH  Google Scholar 

  24. Du FF, Lu JG (2020) New criterion for finite-time stability of fractional delay systems. Appl Math Lett 104:106248

    MathSciNet  MATH  Google Scholar 

  25. Du FF, Lu JG (2020) New criteria on finite-time stability of fractional-order Hopfield neural networks with time delays. IEEE Trans Neural Netw Learn Syst 32:1–9

    MathSciNet  Google Scholar 

  26. Ruan S, Filfil RS (2004) Dynamics of a two-neuron system with discrete and distributed delays. Phys D 191(3–4):323–342

    MathSciNet  MATH  Google Scholar 

  27. Cao JD, Yuan K, Li HX (2006) Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays. IEEE Trans Neural Netw 17(6):1646–1651

    Google Scholar 

  28. Song QK, Yan H, Zhao ZJ, Liu YR (2016) Global exponential stability of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays. Neural Netw 81:1–10

    MATH  Google Scholar 

  29. Tank DW, Hopfield JJ (1987) Neural computation by concentrating information in time. Proc Natl Acad Sci USA 84:1896–1991

    MathSciNet  Google Scholar 

  30. Song YL, Peng YH (2006) Stability and bifurcation analysis on a Logistic model with discrete and distributed delays. Appl Math Comput 181:1745–1757

    MathSciNet  MATH  Google Scholar 

  31. Zhang CH, Xp Yan, Cui Gh (2010) Hopf bifurcations in a predator-prey system with a discrete delay and a distributed delay. Nonlinear Anal Real World Appl 11:4141–4153

    MathSciNet  MATH  Google Scholar 

  32. Celik C, Degirmenci E (2016) Stability and Hopf bifurcation of a predator-prey model with discrete and distributed delays. J Appl Nonlinear Dyn 5(1):73–91

    MathSciNet  MATH  Google Scholar 

  33. Wang XH, Liu HH, Xu CL (2012) Hopf bifurcations in a predator-prey system of population allelopathy with a discrete delay and a distributed delay. Nonlinear Dyn 69:2155–2167

    MathSciNet  MATH  Google Scholar 

  34. Shi RQ, Qi JM, Tang SY (2013) Stability and Hopf bifurcation analysis for a stage-structured predator-prey model with discrete and distributed delays. J Appl Math. https://doi.org/10.1155/2013/201936

    Article  MathSciNet  MATH  Google Scholar 

  35. Dai Y, Zhao H, Lin Y (2012) Stability and Hopf bifurcation analysis on a partial dependent predator-prey system with discrete and distributed delays. In: Chaos-Fractals Theories and Applications (IWCFTA), 2012 Fifth International Workshop on. IEEE, 2012

  36. Wang Z, Liu Y, Liu X (2005) On global asymptotic stability of neural networks with discrete and distributed delays. Phys Lett A 345(4–6):299–308

    MATH  Google Scholar 

  37. Park JH (2007) An analysis of global robust stability of uncertain cellular neural networks with discrete and distributed delays. Chaos Solitons Fractals 32(2):800–807

    MathSciNet  MATH  Google Scholar 

  38. Wang HM, Wei GL, Wen SP, Huang TW (2020) Generalized norm for existence, uniqueness and stability of Hopfield neural networks with discrete and distributed delays. Neural Netw 128:288–293

    MATH  Google Scholar 

  39. Yang XY, Li XD (2018) Finite-time stability of linear non-autonomous systems with time-varying delays. Adv Differ Equ 2018:101

    MathSciNet  MATH  Google Scholar 

  40. Zhang H, Ye R, Liu S, Cao J, Alsaedi A, Li X (2018) LMI based approach to stability analysis for fractional-order neural networks with discrete and distributed delays. Internat J System Sci 49(3):537–545

    MathSciNet  MATH  Google Scholar 

  41. Tyagi S, Abbas S, Hafayed M (2016) Global Mittag-Leffler stability of complex-valued fractional-order neural network with discrete and distributed delays. Rend Circ Mat Palermo 65(3):485–505

    MathSciNet  MATH  Google Scholar 

  42. Srivastava HM, Abbas S, Tyagi S, Lassoued D (2018) Global exponential stability of fractional-order impulsive neural network with time-varying and distributed delay. Math Meth Appl Sci 41(5):2095–2104

    MathSciNet  MATH  Google Scholar 

  43. Song QK, Zhao ZJ, Liu YR (2015) Impulsive effects on stability of discrete-time complex-valued neural networks with both discrete and distributed time-varying delays. Neurocomputing 168:1044–1050

    Google Scholar 

  44. Wu HQ, Zhang XX, Xue SH, Niu PF (2017) Quasi-uniform stability of Caputo-type fractional-order neural networks with mixed delay. Int J Mach Learn Cyb 8(5):1501–1511

    Google Scholar 

  45. Corduneanu C (1971) Principles of differential and intergral equations. Allyn and Bacon, USA

    MATH  Google Scholar 

  46. Yang ZY, Zhang J, Niu YQ (2020) Finite-time stability of fractional-order bidirectional associative memory neural networks with mixed time-varying delays. J Appl Math Comput 63:501–522

    MathSciNet  MATH  Google Scholar 

  47. Millett D (1964) Nonlinear vector integral equations as contraction mappings. Arch Ration Mech Anal 15:79–86

    MathSciNet  Google Scholar 

  48. Naifar O, Jmal A, Nagy AM, Ben MA (2020) Improved quasiuniform stability for fractional order neural nets with mixed delay. Math Probl Eng 2020:1–7

    MathSciNet  MATH  Google Scholar 

  49. Du FF, Lu JG (2021) New approach to finite-time stability for fractional-order BAM neural networks with discrete and distributed delays. Chaos Solitons Fractals 151:111225

    MathSciNet  MATH  Google Scholar 

  50. Du FF, Lu JG (2023) Improved quasi-uniform stability criterion of fractional-order neural networks with discrete and distributed delays. Asian J Control 25:229–240

    MathSciNet  Google Scholar 

  51. Bainov D, Simeonov P (1992) Integral inequalities and applications. Springer, New York

    MATH  Google Scholar 

  52. Kuczma M (2009) An introduction to the theory of functional equations and inequalities: Cauthy’s equation and Jensen’s inequality. Birkh\(\ddot{a}\)user, Boston

  53. Kim YH (2009) Gronwall, Bellman and Pachpatte type integral inequalities with applications. Nonlinear Anal 71:e2641–e2656

    MathSciNet  MATH  Google Scholar 

  54. Bellman R (1943) The stability of solutions of linear differential equations. Duke Math J 10:643–647

    MathSciNet  MATH  Google Scholar 

  55. Syed AM, Usha M, Zhu QX, Shanmugam S (2020) Synchronization analysis for stochastic TS fuzzy complex networks with Markovian jumping parameters and mixed time-varying delays via impulsive control. Math Probl Eng 2020:9739876

    MathSciNet  MATH  Google Scholar 

  56. Syed AM, Yogambigai J, Saravanan S, Elakkia S (2019) Stochastic stability of neutral-type Markovian-jumping BAM neural networks with time varying delays. J Comput Appl Math 349:142–156

    MathSciNet  MATH  Google Scholar 

  57. Syed AM, Balasubramaniam P (2009) Exponential stability of uncertain stochastic fuzzy BAM neural networks with time-varying delays. Neurocomputing 72(4–6):1347–1354

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of Hubei Province (Grant No. 2020CFB637), the Natural Science Foundation of China (Grant No. 61876192), and the Fundamental Research Funds for the of South-Central University for Nationalities (Grant Nos. KTZ20051, CZT20020, YZZ19004).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, South-Central Minzu University, Wuhan, 430074, Hubei, China

    Zhanying Yang, Junhao Hu & Jun Mei

  2. School of Mathematics, Physics and Optoelectronic Engineering, Hubei University of Automotive Technology, Shiyan, 442002, Hubei, China

    Jie Zhang

Authors
  1. Zhanying Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jie Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Junhao Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jun Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

ZY and JH wrote the main manuscript text. JZ and JM prepared figures 1–4 and tables 1–4. All authors reviewed the manuscript.

Corresponding author

Correspondence to Zhanying Yang.

Ethics declarations

Conflict of interest

The authors declare that no potential conflict of interest to be reported to this work.

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 (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

Yang, Z., Zhang, J., Hu, J. et al. Some New Gronwall-Type Integral Inequalities and their Applications to Finite-Time Stability of Fractional-Order Neural Networks with Hybrid Delays. Neural Process Lett 55, 11233–11258 (2023). https://doi.org/10.1007/s11063-023-11373-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-023-11373-3

Keywords

Search

Navigation