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On the comparison of a Kantorovich-type and Moore theorems

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Abstract

It was shown in Zhue and Wolfe (Nonlinear Anal. 15(2):229–232, 1990) that the hypotheses of the affine invariant Moore theorem for solving nonlinear equations using Newton’s method are always valid when those of the Kantorovich theorem due to Deuflhard and Heindl (SIAM J. Numer. Anal. 16:1–10, 1980) hold but not necessarily vice versa. Here we show that this result is not true in general for a weaker version of the Kantorovich theorem shown recently by us in Argyros (Advances in the Efficiency of Computational Methods and Applications, World Scientific, Singapore, 2000; Int. J. Comput. Math. 80:5, 2002) and Argyros and Szidarovszky (The Theory and Applications of Iteration Methods, CRC Press, Boca Raton, 1993).

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References

  1. Argyros, I.K.: Advances in the Efficiency of Computational Methods and Applications. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  2. Argyros, I.K.: On the convergence and application of Newton’s method under weak Hölder continuity assumptions. Int. J. Comput. Math. 80, 5 (2002)

    Google Scholar 

  3. Argyros, I.K., Szidarovszky, F.: The Theory and Applications of Iteration Methods. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

  4. Deuflhard, P., Heindl, G.: Affine invariant convergence theorems for Newton’s method and extensions to related methods. SIAM J. Numer. Anal. 16, 1–10 (1980)

    Article  MathSciNet  Google Scholar 

  5. Kantorovich, L.V., Akilov, G.P.: Functional Analysis in Normed Spaces. Pergamon, Elmsford (1982)

    Google Scholar 

  6. Moore, R.E.: A test for existence of solutions to nonlinear systems. SIAM J. Numer. Anal. 14, 611–615 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  7. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    MATH  Google Scholar 

  8. Neumaier, A., Shen, Z.: The Krawczyk operator and Kantorovich’s theorem. J. Math. Anal. Appl. 149, 437–443 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Rall, L.B.: A comparison of the existence theorems of Kantorovich and Moore. SIAM J. Numer. Anal. 17, 148–161 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Zhue, S., Wolfe, M.A.: A note on the comparison of the Kantorovich and Moore theorems. Nonlinear Anal. 15, 3 (1990), 229–232

    Google Scholar 

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  1. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

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  1. Ioannis K. Argyros
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Correspondence to Ioannis K. Argyros.

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Argyros, I.K. On the comparison of a Kantorovich-type and Moore theorems. J. Appl. Math. Comput. 29, 117–123 (2009). https://doi.org/10.1007/s12190-008-0102-z

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  • DOI: https://doi.org/10.1007/s12190-008-0102-z

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