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The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map. (English) Zbl 1045.47057
In this paper, it is shown that the convergence of Mann iterations is equivalent to the convergence of Ishikawa iterations for asymptotically nonexpansive and asymptotically pseudocontractive mappings, using essentially the technique of {\it B. E. Rhoades} and {\it Ş. M. Şoltuz} [Int. J. Math. Math. Sci. 2003, 2645--2651 (2003; Zbl 1045.47058), see the following review]. In a similar fashion, one can show that the converence of Mann-Ishikawa iterations is equivalent to the convergence of three-step (Noor) iterations [see {\it M. Aslam Noor}, J. Math. Anal. Appl. 251, 217--229 (2000; Zbl 0964.49007)].

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties
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##### References:
 [1] Ishikawa, S.: Fixed points by a new iteration method. Proc. amer. Math. soc. 44, 147-150 (1974) · Zbl 0286.47036 [2] Kato, T.: Nonlinear semigroup and evolution equations. J. math. Soc. Japan 19, 508-520 (1967) · Zbl 0163.38303 [3] Mann, W. R.: Mean value in iteration. Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603 [4] Rhoades, B. E.; Şoltuz, Ş.M: On the equivalence of Mann and Ishikawa iteration methods. Internat. J. Math. math. Sci. 33, 451-459 (2003) · Zbl 1014.47052 [5] Schu, J.: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. math. Anal. appl. 158, 407-413 (1991) · Zbl 0734.47036 [6] Weng, X.: Fixed point iteration for local strictly pseudocontractive mapping. Proc. amer. Math. soc. 113, 727-731 (1991) · Zbl 0734.47042