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Dynamical Systems Method of gradient type for solving nonlinear equations with monotone operators

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Abstract

A version of the Dynamical Systems Method (DSM) of gradient type for solving equation F(u)=f where F:H→H is a monotone Fréchet differentiable operator in a Hilbert space H is studied in this paper. A discrepancy principle is proposed and the convergence to the minimal-norm solution is justified. Based on the DSM an iterative scheme is formulated and the convergence of this scheme to the minimal-norm solution is proved.

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References

  1. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    MATH  Google Scholar 

  2. Hoang, N.S., Ramm, A.G.: Solving ill-conditioned linear algebraic systems by the dynamical systems method. Inverse Probl. Sci. Eng. 16(5), 617–630 (2008)

    Article  MathSciNet  Google Scholar 

  3. Hoang, N.S., Ramm, A.G.: An iterative scheme for solving nonlinear equations with monotone operators. BIT Numer. Math. 48(4), 725–741 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hoang, N.S., Ramm, A.G.: A discrepancy principle for equations with monotone continuous operators. Nonlinear Anal. Ser. A, Theory Methods Appl. 70, 4307–4315 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hoang, N.S., Ramm, A.G.: Dynamical systems Gradient method for solving nonlinear equations with monotone operators. Acta Appl. Math. 106, 473–499 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hoang, N.S., Ramm, A.G.: Dynamical systems method for solving nonlinear equations with monotone operators. Math. Comput. 79, 239–258 (2010)

    MATH  MathSciNet  Google Scholar 

  7. Ivanov, V., Tanana, V., Vasin, V.: Theory of Ill-Posed Problems. VSP, Utrecht (2002)

    MATH  Google Scholar 

  8. Jin, Q., Tautenhahn, U.: On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems. Numer. Math. 111(4), 509–558 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. de Gruyter, Berlin (2008)

    Book  MATH  Google Scholar 

  10. Lions, J.L.: Quelques Methodes de Resolution des Problemes aux Limites non Lineaires. Dunod, Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  11. Morozov, V.A.: Methods of Solving Incorrectly Posed problems. Springer, New York (1984)

    Google Scholar 

  12. Pascali, D., Sburlan, S.: Nonlinear Mappings of Monotone Type. Noordhoff, Leyden (1978)

    MATH  Google Scholar 

  13. Ramm, A.G.: Dynamical systems method for solving nonlinear operator equations. Int. J. Appl. Math. Sci. 1(1), 97–110 (2004)

    MATH  Google Scholar 

  14. Ramm, A.G.: Dynamical systems method (DSM) for unbounded operators. Proc. Am. Math. Soc. 134(4), 1059–1063 (2006)

    Article  MATH  Google Scholar 

  15. Ramm, A.G.: Dynamical Systems Method for Solving Operator Equations. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  16. Skrypnik, I.V.: Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. Am. Math. Soc., Providence (1994)

    MATH  Google Scholar 

  17. Tautenhahn, U.: On the method of Lavrentiev regularization for nonlinear ill-posed problems. Inverse Probl. 18, 191–207 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Vainberg, M.M.: Variational Methods and Method of Monotone Operators in the Theory of Nonlinear Equations. Wiley, New York (1973)

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Kansas State University, Manhattan, KS, 66506-2602, USA

    N. S. Hoang

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  1. N. S. Hoang
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Correspondence to N. S. Hoang.

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Communicated by Erkki Somersalo.

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Hoang, N.S. Dynamical Systems Method of gradient type for solving nonlinear equations with monotone operators. Bit Numer Math 50, 751–780 (2010). https://doi.org/10.1007/s10543-010-0284-2

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  • DOI: https://doi.org/10.1007/s10543-010-0284-2

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