# Article

Keywords:
nonlinear operator equations; Banach spaces; Halley type method; Ostrowski-Kantorovich convergence theorem; Ostrowski-Kantorovich assumptions; optimal error \linebreak bound; \$S\$-order of convergence; sufficient asymptotic error bound
Summary:
Ostrowski-Kantorovich theorem of Halley method for solving nonlinear operator equations in Banach spaces is presented. The complete expression of an upper bound for the method is given based on the initial information. Also some properties of \$S\$-order of convergence and sufficient asymptotic error bound will be discussed.
References:
 Dong Chen: On a New Definition of Order of Convergence in General Iterative Methods I: One Point Iterations. Research Report No. 7, Department of Mathematical Sciences, University of Arkansas, 1991.
 Dong Chen: On a New Definition of Order of Convergence in General Iterative Methods II: Multipoint Iterations. Research Report No. 8, Department of Mathematical Sciences, University of Arkansas, 1991.
 Kantorovich L.V., Akilov G.P.: Functional Analysis in Normed Spaces. Pergaman Press, New York, 1964. MR 0213845 | Zbl 0127.06104
 Ostrowski A.M.: Solution of Equations in Euclidean and Banach Spaces. Academic Press, New York, 3rd ed., 1973. MR 0359306 | Zbl 0304.65002
 Taylor A.E.: Introduction to Functional Analysis. Wiley, New York, 1957. MR 0098966 | Zbl 0654.46002
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