We study the approximation of zero for sum of accretive operators using a modified Mann type forward-backward splitting algorithm and obtain strong convergence of the sequence generated by our scheme to the zero of sum of accretive operators in uniformly convex real Banach spaces which are also uniformly smooth. Our result is new and complements many recent and important results in this direction in the literature.

Let

The normalized duality mapping

Let

Let

It is well known that a forward-backward splitting method (please see, e.g., [

We note that for

Recently, Wei and Duan [

Let

We observe that Theorem

Can we obtain strong convergence result for approximation of zero of sum of accretive operators in

Our interest in this paper is to answer the above question in the affirmative. That is, we establish strong convergence result for approximation of zero of sum of accretive operators in uniformly smooth Banach spaces which are also uniformly convex without assuming that the normalized duality mapping

In the sequel, we will assume that

In what follows we will make use of the following lemmas.

Let

For every

Assume

Let

Let

Let

for

for

Let

In a real Hilbert space, we see that

In the result of Suantai and Cholamjiak [

What uniformly smooth Banach spaces (except Hilbert spaces) satisfy the assumption

In

In our more general setting, throughout this paper, we will assume that

By following the same method of proof as contained in the proof of Lemma

Let

We will adopt the following notation in this paper:

We first prove the following lemma in a real Banach space with Fréchet differentiable norm.

Let

For each

Using the idea of the proof partly taken from [

Let

Let

If the mapping

Let

Recall that a mapping

Let

The following lemma will be used in the next theorem.

Let

We now consider the viscosity approximation method with weakly contractive mapping in the following theorem.

Let

We first observe that

All the results in this paper can still be obtained if error terms are added to our iterative scheme (

We make the following comments which highlight our contributions in this paper (please see also [

The results of Wei and Shi [

The condition

for all

Our results extend the results of [

In our result in this paper, we consider the viscosity approximation method with weakly contractive mapping for solving (

We specifically clarify the differences between the results of this paper and the results obtained in [

We state here that the results obtained in Wei and Duan [

The choice of sequence

does not satisfy the extra condition

The author declares that there is no conflict of interests regarding the publication of this paper.