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In this paper, we introduce the modified iterations of Mann's type for nonexpansive mapping and asymptotically nonexpansive mapping to have the strong and weak convergence in a uniformly convex Banach space.

In 2005, Kim and Xu [16] modified Mann's iteration scheme and the modified iteration method still works in a Banach space.

A modified iteration procedure implemented in this work gives convergence of the solution after three iterations.

Very recently, Şahin and Başarir [7] modified iteration process (1.6) in a (operatorname{CAT}(0)) space as follows.

Also (6) reduces to (3) when T n = T for all n ≥ 1. Very recently, Şahin and Başarir [10] modified iteration process (6) in a CAT ( 0 ) space as follows.

Iteration scheme (2.6) is independent of modified Noor iteration, modified Ishikawa iteration and modified Mann iteration schemes.

This iteration is independent of those of modified Mann iteration in [31] and modified Ishikawa iteration in [35] and reduces to S-iteration of Agarwal et al. [2] when (T^{n}=T) for all (nin mathbb {N}).

The contraction is replaced by the generalized contraction in both modified Mann iterations (1.8) and (1.9).

(ii) The contraction is replaced by the generalized contraction in both modified Mann iterations (1.8) and (1.9).

The following proposition is our substitute for the use of Banach limits in convergence proofs of modified Halpern iterations.

We will call these iterations secondary modified Halpern iterations.