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An application of Newton's method to differential and integral equations

Published online by Cambridge University Press:  17 February 2009

J. M. Gutiérrez  and
M. A. Hernández
J. M. Gutiérrez
Affiliation:
Department of Mathematics and Computation, University of La Rioja, C/ Luis de Ulloa s/n, 26004, Logroño, Spain.
M. A. Hernández
Affiliation:
Department of Mathematics and Computation, University of La Rioja, C/ Luis de Ulloa s/n, 26004, Logroño, Spain.
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Abstract

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Newton's method is applied to an operator that satisfies stronger conditions than those of Kantorovich. Convergence and error estimates are compared in the two situations. As an application, we obtain information on the existence and uniqueness of a solution for differential and integral equations.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 3 , January 2001 , pp. 372 - 386
Copyright
Copyright © Australian Mathematical Society 2001

References

[1] Argyros, I. K., “Newton-like methods under mild differentiability conditions with error analysis”. Bull. Austr. Math. Soc. 37 (1988) 131–147.CrossRefGoogle Scholar
[2] Braun, M., Differential equations and their applications (Springer, 1983).Google Scholar
[3] Gutiérrez, J. M., “A new semilocal convergence theorem for Newton's method”, J. Comput. Appl. Math. 79 (1997) 131–145.CrossRefGoogle Scholar
[4] Gutiérrez, J. M., Hernández, M. A. and Salanova, M. A., “Accessibility of solutions by Newton's method”. Inter. J. Computer Math. 57 (1995) 239–247.CrossRefGoogle Scholar
[5] Huang, Z., “A note on the Kantorovich theorem for Newton iteration”, J. Comput. Appl. Math. 47 (1993) 211–217.Google Scholar
[6] Kantorovich, L. V. and Akilov, G. P., Functional analysis (Pergamon Press, 1982).Google Scholar
[7] Ostrowski, A. M., Solution of equations in Euclidean and Banach spaces (Academic Press, 1943).Google Scholar
[8] Porter, D. and Stirling, D. S. G., Integral equations, a practical treatment, from spectral theory to applications (Cambridge, 1990).CrossRefGoogle Scholar
[9] Rall, L. B., Computational solution of nonlinear operator equations (Robert E. Krieger Publishing Company, 1979).Google Scholar
[10] Rheinboldt, W. C., “A unified convergence theory for a class of iterative processes”, SIAM J. Numer. Anal. 5 (1968) 42–43.CrossRefGoogle Scholar
[11] Rockne, J., “Newton's method under mild differentiability conditions with error analysis”, Numer. Math. 18 (1972) 401–412.CrossRefGoogle Scholar
[12] Simmons, F., Differential equations with applications and historical notes (McGraw-Hill, 1972).Google Scholar
[13] Yamamoto, T., “A convergence theorem for Newton-like methods in Banach spaces”, Numer. Math. 51 (1987) 545–557.CrossRefGoogle Scholar
[14] Yamamoto, T., “On the method of tangent hyperbolas in Banach spaces”, J. Comput. Appl. Math. 21 (1988) 75–86.CrossRefGoogle Scholar