REMARKS ON THE CONVERGENCE OF NEWTON'S METHOD UNDER HÖLDER CONTINUITY CONDITIONS

Authors

  • IOANNIS K. ARGYROS Department of Mathematics, Cameron University, Lawton, OK, 73505, U.S. A.

DOI:

https://doi.org/10.5556/j.tkjm.23.1992.4550

Keywords:

Newton-like iteration, Banach space, Fréchet-derivative

Abstract

We use a Newton-like iteration to solve the nonlinear op­ erator equation in a Banach space. The basic assumption is that the Fréchet-derivative of the nonlinear operator is Hölder continuous on some open ball centered at the initial guess. Under natural assumptions, we prove linear convergence of the iteration to a locally unique solution of the nonlinear equation.

References

L. M. Graves, "Some mapping theorems", Duke Math. J., Vol 17 (1950), pp. 111-114.

L. V. Kantorovich, "On Newton's method", Math. Reviews, Vol. 12 (1951), p. 419.

L. V. Kantorovich and G. P. Akilov, "Functional analysis in normed spaces", Oxford Publ., Pergamon Press, 1964.

I. P. Mysovkih, "On the convergence of Newton's method", Math. Reviews, Vol. 12 (1951), p.419.

L. B. Rall, "Nonlinear functional analysis and applications", (Article by J. Dennis.) Academic Press, 1971.

W. C. Rheinholdt, "Numerical analysis of parametrized nonlinear equations", John Wiley, Publ. 1986.

L. M. Stein, "Sufficient conditions for the convergence of Newton's method in complex Banach spaces", Proc. Amer. Math. Soc. Vol. 3 (1952), pp. 858-863.

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Published

1992-12-01

How to Cite

ARGYROS, I. K. (1992). REMARKS ON THE CONVERGENCE OF NEWTON’S METHOD UNDER HÖLDER CONTINUITY CONDITIONS. Tamkang Journal of Mathematics, 23(4), 269–277. https://doi.org/10.5556/j.tkjm.23.1992.4550

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