Abstract
The iteration maps of Euler family for finding zeros of an operatorf in Banach spaces is defined as the partial sum of Taylor expansion of the local inversef -1 z off atz. The unified convergence theorem is established for the iterations of Euler family under the assumption that\(\alpha \leqslant 3 - 2\sqrt 2 \), while the strong condition thatf is analytic in Smale’s criterion α is replaced by the weak condition thatf is of finite order derivative.
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Wang, X. Convergence of iterations of Euler family under weak condition. Sci. China Ser. A-Math. 43, 958–962 (2000). https://doi.org/10.1007/BF02879801
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DOI: https://doi.org/10.1007/BF02879801