A new convergence theorem for the method of tangent hyperbolas in banach space

Ioannis K. Argyros


In this study we appmximate a locally unique solution of a non-linear operator equtation in Banach space using the method of tangent hyperbolas. A new semilocal convergence theorem is provided using Lipschitz conditions on the second Fréchet-derivative. Our conditions are different than earlier ones. Hence, they have theorctical and practical value. Numerical examples are also provoded.

Palabras clave

Banach space ; Method of tangent hyperbolas ; Fréchet - derivative, Newton - Kantorovich hypothesis

Texto completo:



M. Altman, Concerning the method of tangent hyperbolas for operator equations, Bull. Acad. Polon. Sci. Ser. Math. Astr-. Phys. 9, pp. 633- 637, (1961).

I.K. Argyros, On the convergence of an Euler-Chebysheff-type method under Newton-Kantorovich-type hypotheses, Pure Mathematics and Applications, 4 3, pp. 369- 373, (1993).

I.K. Argyros, On the method of tangent hyperbolas, J. Appmx. Th. Appl. 12 1, pp. 78- 96, (1996).

I.K. Argyros, Polynomial Operator- Equations in Abstract Spaces and Applications, CRC Press LLC, Boca Raton, Florida, (1998).

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

J.A. Ezquerro, J.M. Gutierez, and M.A. Hernandez, A construction procedure of iterative methods with cubical convergence, Appl. Math. Comp. (to appear).

L.M. Graves, Riemann integration and Taylor's theorem in general analysis, Trans. Amer. Math. Soc. 29, pp. 163- 177, (1927).

J.M. Gutierez, A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math. 79 (1997), 131-145.

S. Kanno, Convergence theorems for the method of tangent hyperbolas, Afath. Japanich, 37, 4, pp. 711- 722, (1992).

L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, (1982).

R.A. Safiev, The method of tangent hyperbolas, Sov. Math. Dokl. 4, pp. 482- 485, (1963).

DOI: http://dx.doi.org/10.22199/S07160917.1999.0001.00001

Enlaces refback

  • No hay ningún enlace refback.