Skip to main content
SpringerLink
Log in

On a general nonlinear problem with distributed delays

  • Differential Equations
  • Published:
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) Aims and scope Submit manuscript

Abstract

The paper considers a general system of ordinary differential equations appearing in the neural network theory. The activation functions are assumed to be continuous and bounded by power type functions of the states and distributed delay terms. These activation functions are not necessarily Lipschitz continuous as it is commonly assumed in the literature. We obtain sufficient conditions for exponential decay of solutions.

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

  1. G. Bao and Z. Zeng, “Analysis and design of associative memories based on recurrent neural network with discontinuous activation functions”, Neurocomputing, 77, 101–107, 2012.

    Article  Google Scholar 

  2. Z. Cai and L. Huang, “Existence and global asymptotic stability of periodic solution for discrete and distributed time-varying delayed neural networks with discontinuous activations”, Neurocomputing, 74, 3170–3179, 2011.

    Article  Google Scholar 

  3. U. D. Dongade and S. G. Deo, “Pointwise estimates of solutions of some Volterra integral equations”, J. Math. Anal. Appl., 45 (3), 615–628, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Forti, M. Grazzini, P. Nistri and L. Pancioni, “Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations”, PhysicaD, 214, 88–99, 2006.

    MathSciNet  MATH  Google Scholar 

  5. B. Kosko, Neural Network and Fuzzy System - A Dynamical System Approach toMachine Intelligence (Prentice-Hall of India, New Delhi, 1991).

    Google Scholar 

  6. V. Lakshmikhantam and S. Leela, Differential and Integral Inequalities: Theory and Applications, Mathematics in Sciences and Engineering, (Acad. Press, New York-London, 1969).

    Google Scholar 

  7. S. Mohamad, K. Gopalsamy and H. Akca, “Exponential stability of artificial neural networks with distributed delays and large impulses”, Nonl. Anal.: RealWorld Applications, 9, 872–888, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Park, “On global stability criterion of neural networks with continuously distributed delays”, Chaos Solitons & Fractals, 37, 444–449, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  9. Z. Qiang, M. A. Run-Nian and X. Jin, “Global exponential convergence analysis ofHopfieldNeuralNetworks with continuously distributed delays”, Commun. Theor. Phys., 39 (3), 381–384, 2003.

    Article  MathSciNet  Google Scholar 

  10. N.-e. Tatar, “Hopfield neural networkswith unbounded monotone activation functions”, Adv. Artificial Neural Netw. Syst., ID 571358, 1–5, 2012.

    Google Scholar 

  11. N.-e. Tatar, “Control of systems with Hölder continuous functions in the distributed delays”, Carpathian J. Math., 30 (1), 123–128, 2014.

    MathSciNet  MATH  Google Scholar 

  12. Y. X. Wang, Q. Y. Zhou, B. Xiao and Y. H. Yu, “Global exponential stability of cellular neural networks with continuously distributed delays and impulses”, Physics Letters A, 350, 89–95, 2006.

    Article  MATH  Google Scholar 

  13. H. Wu, F. Tao, L. Qin, R. Shi and L. He, “Robust exponential stability for interval neural networks with delays and non-Lipschitz activation functions”, Nonlinear Dyn., 66, 479–487, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  14. Q. Zhang, X. P. Wei and J. Xu, “Global exponential stability of Hopfield neural networks with continuously distributed delays”, Physics Letters A, 315, 431–436, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Zhou, S. Y. Li and Z. G. Yang, “Global exponential stability of Hopfield neural networks with distributed delays”, Appl.Math. Model., 33, 1513–152, 2009.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    N. Tatar

Authors
  1. N. Tatar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Tatar.

Additional information

Original Russian Text © N. Tatar, 2017, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2017, No. 4, pp. 72-80.

The author is grateful for the financial support and the facilities provided by King Abdulaziz City of Science and Technology (KACST) under the National Science, Technology and Innovation Plan (NSTIP), Project No. 15-OIL4884-0124.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tatar, N. On a general nonlinear problem with distributed delays. J. Contemp. Mathemat. Anal. 52, 184–190 (2017). https://doi.org/10.3103/S1068362317040045

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1068362317040045

Keywords

Search

Navigation