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On the Rate of Convergence of Gradient Flow for Some Evolution Systems

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Abstract

The rate of convergence of first- and second-order evolution equations governed by the subdifferential of a proper, convex, and lower semicontinuous function has been studied.

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References

  1. Aubin, J.P., Ekland, I.: Applied Nonlinear Analysis. Pure and Applied Mathematics. Wiley, New York (1984). A Wiley-Interscience Publication

    MATH  Google Scholar 

  2. Morosanu, G.: Nonlinear Evolution Equations and Applications. Mathematics and its Applications, vol. 26. Reidel, Dordrecht (1988)

    MATH  Google Scholar 

  3. Apreutesei, N.C.: Nonlinear Second Order Evolution Equation of Monotone Type. Pushpa Publishing House, Allahabad (2007)

    Google Scholar 

  4. Bruck, R.E.: Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal. 18, 15–26 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baillon, J.B.: Un exemple concernant le comportement asymptotique de la solution du problème du/dt+∂φ(u)∋0. J. Funct. Anal. 28, 369–376 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Okochi, H.: A note on asymptotic strong convergence of nonlinear contraction semigroups. Proc. Jpn. Acad., Ser. A, Math. Sci. 56, 83–84 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Güler, O.: Convergence rate estimates for the gradient differential inclusion. Optim. Methods Softw. 20, 729–735 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barbu, V.: A class of boundary problems for second order abstract differential equation. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 19, 295–319 (1972)

    MathSciNet  MATH  Google Scholar 

  9. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leiden (1976)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Djafari Rouhani, B., Khatibzadeh, H.: Asymptotic behavior of solutions to some homogeneous second order evolution equations of monotone type. J. Inequal. Appl. (2007)

  14. Djafari Rouhani, B., Khatibzadeh, H.: Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations. Nonlinear Anal. 70, 4369–4376 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Djafari Rouhani, B., Khatibzadeh, H.: Asymptotic behavior of bounded solutions to a nonhomogeneous second order evolution equation of monotone type. Nonlinear Anal. 71, e147–e152 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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  MATH  Google Scholar 

  17. Djafari Rouhani, B., Khatibzadeh, H.: A strong convergence theorem for solutions to a nonhomogeneous second order evolution equations. J. Math. Anal. Appl. 363, 648–654 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Véron, L.: Un exemple concernant le comportement asymptotique de la solution bornée de l’équation \(\frac{d^{2}u}{dt^{2}} \in \partial \varphi (u)\). Monatshefte Math. 89, 57–67 (1980)

    Article  MATH  Google Scholar 

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Acknowledgements

This research was in part supported by a grant from university of Zanjan (No. 9041). The author would like to thank the referees for valuable comments.

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Correspondence to Hadi Khatibzadeh.

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Communicated by Viorel Barbu.

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Khatibzadeh, H. On the Rate of Convergence of Gradient Flow for Some Evolution Systems. J Optim Theory Appl 154, 685–690 (2012). https://doi.org/10.1007/s10957-012-0020-y

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