We prove the equivalence and the strong convergence of iterative processes involving generalized strongly asymptotically

Throughout this paper, we assume that

In 1967, Browder [

In 1972, Goebel and Kirk [

Let

This class is more general than the class of nonexpansive mappings as the following example clearly shows.

If

In 1974, Deimling [

An operator

Note that in the special case in which

Osilike [

Since an operator

Recently, has been also studied the following class of maps.

A mapping

Choosing

In 1991, Schu [

Let

Obviously every asymptotically nonexpansive mapping is asymptotically pseudocontractive, but the converse is not valid; it is well known that

In [

Let

Until 2009, no results on fixed-point theorems for asymptotically pseudocontractive mappings have been proved. First, Zhou in [

In this paper, our attention is on the class of the

If

One can note that

if

the mapping

We study the equivalence between three kinds of iterative methods involving the generalized asymptotically strongly

Moreover, we prove that these methods are equivalent and strongly convergent to the unique fixed point of the generalized strongly asymptotically

We will briefly introduce some of the results in the same line of ours. In 2001, [

In 2003, Chidume and Zegeye [

Taking in to account Chidume and Zegeye [

In particular, if

We also introduce an implicit iterative process as follows:

The algorithm is well defined. Indeed, if

These kind of iterative processes (also called by Chang

In [

In the next section, we prove that, in the setting of the uniformly smooth Banach space, if

We recall some definitions and conclusions.

Let

The next lemma is one of the main tools for our proofs.

Let

The proof is the same as in [

Let

if

if one has

If in Lemma

The ideas of the proofs of our main Theorems take in to account the papers of Chang and Chidume et al. [

Let

Let

the modified Ishikawa iteration sequence with errors (

the modified Mann iteration sequence with errors (

First of all, we note that by boundedness of the range of

In this case, we can prove that

Firstly, we note that substituting (

Moreover, we observe that

Take

We prove, by induction, that

By (

From (

If there are only finite indices for which

Let

Let

Suppose moreover that the sequences

the modified Mann iteration sequence with errors (

the implicit iteration sequence with errors (

As in Theorem

the set of indices for which

the set of indices for which

We can prove that

Let

In (

Let

Let

Firstly, we observe that, by the boundedness of the range of

By Lemma

Now, we will prove that the sequence

By (

In the same manner, by induction, one obtains that, for every

Let

the modified Ishikawa iteration sequence with errors (

the modified Mann iteration sequence with errors (

the implicit iteration sequence with errors (