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

$\mathit{\Phi}$-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

$\mathit{\Phi}$-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)].