We present a new iterative scheme with errors to solve the problems of finding common zeros of finite

Let

Let

A mapping

A mapping

A subset

We use

A mapping

Interest in accretive mappings, which is an important class of nonlinear operators, stems mainly from their firm connection with equations of evolution. It is well known that many physically significant problems can be modelled by initial value problems of the form

Recall that the following normal Mann iterative scheme to approximate the fixed point of a nonexpansive mapping

Later, many mathematicians try to combine the ideas of proximal method and Mann iterative method to approximate the zeros of

In particular, in 2007, Qin and Su [

Motivated by iterative schemes (

The work in [

In 2009, Yao et al. presented the following iterative scheme in the frame of Hilbert space in [

Then

Motivated by the work in [

Let

Inspired by the work in [

Note that there are some differences between our work and Shehu and Ezeora’s in [

The idea of three-step iteration is employed in our paper.

The error sequence

Recall that, in [

In Section

Next, we list some results we need in sequel.

Let

Let

Let

In a real Banach space

Let

Let

Let

Let

Let

Let

Let

The main idea of the proof is essentially from that in [

It is easy to check that

On the other hand, for all

For all

Then

Therefore,

Let

From Lemma

For all

For all

Then repeating the discussion in Lemma

Let

Then

We will split the proof into five steps.

We will first show that

By using the induction method, we see that, for

Suppose that (

Thus (

For all

Since

Then we set

In fact,

Next, we discuss

If

If

Combining (

Putting (

Similarly, we have

Therefore, from (

From Step 1, we know that

In fact,

Noticing the results of Steps 1 and 2 and

Since

From Lemma

Since

So

Since

Moreover, noticing the fact that

Since

Let

Let

From (

This completes the proof.

If, in Theorem

Let

Let

Then

In what follows in this paper, unless otherwise stated, we will assume that

Now, we will examine the following nonlinear elliptic systems: