Skip to main content
SpringerLink
Log in

Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We consider a second-order differential equation governed by a quasiconvex function with a nonempty set of minimizers. Assuming that the gradient of this function is Lipschitz continuous, the existence of solutions to the gradient system is guaranteed. We study the asymptotic behavior of these solutions in continuous and discrete times. More precisely, we show that, if a solution is bounded, then it converges weakly to a critical point of the function; otherwise, it goes to infinity (in norm). We also provide several sufficient conditions for obtaining strong convergence in both continuous and discrete cases. Our work is motivated by an open problem proposed by Khatibzadeh and Moroşanu (J Convex Anal 26:1175–1186, 2019), and we solve this problem in the case, where the gradient of the function is Lipschitz continuous on bounded sets.

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

Similar content being viewed by others

Data Availibility

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Barbu, V.: Sur un problème aux limites pour une classe d’équations différentielles non linéaires abstraits du deuxiéme ordre en t. C. R. Acad. Sci. Paris, Sér.A-B 27, A459–A462 (1972)

    MATH  Google Scholar 

  2. Barbu, V.: A class of boundary problems for second order abstract differential equations. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 19, 295–319 (1972)

    MathSciNet  MATH  Google Scholar 

  3. Moroşanu, G.: Asymptotic behaviour of solutions of differential equations associated to monotone operators. Nonlinear Anal. 3, 873–883 (1979)

    Article  MathSciNet  Google Scholar 

  4. Moroşanu, G.: Nonlinear Evolution Equations and Applications. Reidel, Dordrecht (1988)

    MATH  Google Scholar 

  5. Djafari Rouhani, B., Khatibzadeh, H.: Asymptotic behavior of bounded solutions to some second order evolution systems. Rocky Mt. J. Math. 40, 1289–1311 (2010)

    Article  MathSciNet  Google Scholar 

  6. Mitidieri, E.: Asymptotic behaviour of some second order evolution equations. Nonlinear Anal. 6, 1245–1252 (1982)

    Article  MathSciNet  Google Scholar 

  7. Mitidieri, E.: Some remarks on the asymptotic behaviour of the solutions of second order evolution equations. J. Math. Anal. Appl. 107, 211–221 (1985)

    Article  MathSciNet  Google Scholar 

  8. Khatibzadeh, H.: On the rate of convergence of gradient flow for some evolution systems. J. Optim. Theory Appl. 154, 685–690 (2012)

    Article  MathSciNet  Google Scholar 

  9. Brock, W., Xepapadeas, A., Yannacopoulos, A.: Optimal agglomerations in dynamic economics. J. Math. Econom. 53, 1–15 (2014)

    Article  MathSciNet  Google Scholar 

  10. Brock, W., Xepapadeas, A., Yannacopoulos, A.: Spatial externalities and agglomeration in a competitive industry. J. Econom. Dyn. Control 42, 143–174 (2014)

    Article  MathSciNet  Google Scholar 

  11. Khatibzadeh, H., Moroşanu, G.: Asymptotic behavior of solutions to a second order gradient equation of pseudo-convex type. J. Convex Anal. 26, 1175–1186 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Djafari Rouhani, B., Khatibzadeh, H.: A Note on the asymptotic behavior of solutions to a second order difference equation. J. Differ. Equ. Appl. 14, 429–432 (2008)

    Article  MathSciNet  Google Scholar 

  13. Djafari Rouhani, B., Khatibzadeh, H.: Asymptotics of a difference equation and zeroes of monotone operators. Numer. Funct. Anal. Optim. 36, 350–363 (2015)

    Article  MathSciNet  Google Scholar 

  14. Khatibzadeh, H.: On the convergence of solutions to a second order difference equation with monotone operator. Numer. Funct. Anal. Optim. 32, 1271–1282 (2011)

    Article  MathSciNet  Google Scholar 

  15. Mitidieri, E., Moroşanu, G.: Asymptotic behavior of the solutions of second-order difference equations associated to monotone operators. Numer. Funct. Anal. Optim. 8, 419–434 (1985–1986)

  16. Moroşanu, G.: Second order difference equations of monotone type. Numer. Funct. Anal. Optim. 1, 441–450 (1979)

    Article  MathSciNet  Google Scholar 

  17. Poffald, E.I., Reich, S.: An incomplete Cauchy problem. J. Math. Anal. Appl. 113, 514–543 (1986)

    Article  MathSciNet  Google Scholar 

  18. Goudou, X., Munier, J.: The gradient and heavy ball with friction dynamical systems: the quasiconvex case. Math. Program. Ser. B 116, 173–191 (2009)

    Article  MathSciNet  Google Scholar 

  19. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)

    Book  Google Scholar 

  20. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to anonymous reviewers for their careful reading of the manuscript and for their useful and constructive comments and questions leading to the improvement of the paper.

Author information

Authors and Affiliations

  1. Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran

    Mohsen Rahimi Piranfar

  2. University of Zanjan, Zanjan, Iran

    Hadi Khatibzadeh

Authors
  1. Mohsen Rahimi Piranfar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hadi Khatibzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohsen Rahimi Piranfar.

Additional information

Communicated by Nikolai Pavlovich Osmolovskii.

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

Rahimi Piranfar, M., Khatibzadeh, H. Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function. J Optim Theory Appl 188, 169–191 (2021). https://doi.org/10.1007/s10957-020-01784-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-020-01784-w

Keywords

Mathematics Subject Classification

Search

Navigation