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Dynamical systems method for solving nonlinear equations with monotone operators


Authors: N. S. Hoang and A. G. Ramm
Journal: Math. Comp. 79 (2010), 239-258
MSC (2000): Primary 65R30; Secondary 47J05, 47J06, 47J35
DOI: https://doi.org/10.1090/S0025-5718-09-02260-1
Published electronically: April 2, 2009
MathSciNet review: 2552225
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Abstract | References | Similar Articles | Additional Information

Abstract: A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators in a Hilbert space is studied in this paper. An a posteriori stopping rule, based on a discrepancy-type principle is proposed and justified mathematically. The results of two numerical experiments are presented. They show that the proposed version of DSM is numerically efficient. The numerical experiments consist of solving nonlinear integral equations.


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  • 1. K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404 (86j:47001)
  • 2. E. Hairer, and S. P. Norsett, and G. Wanner, Solving ordinary differential equations. I, Nonstiff problems, Springer-Verlag, Berlin, 1993. MR 1227985 (94c:65005)
  • 3. N. S. Hoang and A. G. Ramm, An iterative scheme for solving equations with monotone operators, BIT, 48, no. 4, (2008), 725-741.
  • 4. V. Ivanov, V. Tanana and V. Vasin, Theory of ill-posed problems, VSP, Utrecht, 2002. MR 2010817 (2004j:47020)
  • 5. V. A. Morozov, Methods of solving incorrectly posed problems, Springer-Verlag, New York, 1984. MR 766231 (86d:65005)
  • 6. J. Ortega, W. Rheinboldt, Iterative solution of nonlinear equations in several variables, SIAM, Philadelphia, 2000. MR 1744713 (2000j:65005)
  • 7. D. Pascali and S. Sburlan, Nonlinear mappings of monotone type, Noordhoff, Leyden, 1978. MR 531036 (80g:47056)
  • 8. A. G. Ramm, Dynamical systems method for solving operator equations, Elsevier, Amsterdam, 2007. MR 2281366 (2007m:37220)
  • 9. A. G. Ramm, Global convergence for ill-posed equations with monotone operators: the dynamical systems method, J. Phys A, 36, (2003), L249-L254. MR 1985201 (2004d:47116)
  • 10. A. G. Ramm, Dynamical systems method for solving nonlinear operator equations, International Jour. of Applied Math. Sci., 1, no. 1, (2004), 97-110.
  • 11. A. G. Ramm, Dynamical systems method for solving operator equations, Commun. in Nonlinear Sci. and Numer. Simulation, 9, no. 4, (2004), 383-402. MR 2045643 (2004m:47161)
  • 12. A. G. Ramm, DSM for ill-posed equations with monotone operators, Commun. in Nonlinear Sci. and Numer. Simulation, 10, no. 8, (2005), 935-940. MR 2130199 (2005k:65104)
  • 13. A. G. Ramm, Discrepancy principle for the dynamical systems method, Commun. in Nonlinear Sci. and Numer. Simulation, 10, no. 1, (2005), 95-101. MR 2090273 (2005e:47027)
  • 14. A. G. Ramm, Dynamical systems method (DSM) and nonlinear problems, in the book: Spectral Theory and Nonlinear Analysis, World Scientific Publishers, Singapore, 2005, 201-228. (ed., J. Lopez-Gomez). MR 2328358 (2008d:47143)
  • 15. A. G. Ramm, Dynamical systems method (DSM) for unbounded operators, Proc. Amer. Math. Soc., 134, no. 4, (2006), 1059-1063. MR 2196039 (2006j:47112)
  • 16. M. M. Vainberg, Variational methods and method of monotone operators in the theory of nonlinear equations, Wiley, London, 1973. MR 0467428 (57:7286b)

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Additional Information

N. S. Hoang
Affiliation: Mathematics Department, Kansas State University, Manhattan, Kansas 66506-2602
Email: nguyenhs@math.ksu.edu

A. G. Ramm
Affiliation: Mathematics Department, Kansas State University, Manhattan, Kansas 66506-2602
Email: ramm@math.ksu.edu

DOI: https://doi.org/10.1090/S0025-5718-09-02260-1
Keywords: Dynamical systems method (DSM), nonlinear operator equations, monotone operators, discrepancy principle
Received by editor(s): April 3, 2008
Received by editor(s) in revised form: January 17, 2009
Published electronically: April 2, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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