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The equivalence between the convergences of multi-step iterations with errors for uniformly generalized Lipschitz continuous operators. (Chinese) Zbl 1223.47088
Summary: In this paper, the equivalence of the convergence of modified Mann iterations and multi-step Noor iterations with errors is investigated for uniformly generalized Lipschitz continuous and successively asymptotically $𝛷$-strongly pseudo-contractive type operators in uniformly smooth Banach spaces. Z.-Y. Huang [J. Math. Anal. Appl. 329, No. 2, 935–947 (2007; Zbl 1153.47307)] showed the equivalence of the convergence criteria between modified Mann and Ishikawa iterations with errors for successively $𝛷$-strongly pseudo-contractive operators with bounded range in uniformly smooth Banach spaces. The results obtained in this paper generalize the results of Huang and give an affirmative answer to the conjecture raised by B. E. Rhoades and Ş. M. Şoltuz [Int. J. Math. Math. Sci. No. 7, 451–459 (2003; Zbl 1014.47052)].
MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces