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The purpose of this paper is to present two new forward-backward splitting schemes with relaxations and errors for finding a common element of the set of solutions to the variational inclusion problem with two accretive operators and the set of fixed points of strict pseudocontractions in infinite-dimensional Banach spaces. Under mild conditions, some weak and strong convergence theorems for approximating these common elements are proved. The methods in the paper are novel and different from those in the early and recent literature. Further, we consider the problem of finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a strict pseudocontractions.

The theory of nonexpansive mappings is very important because it is applied to convex optimization, the theory of nonlinear evolution equations, and others. Browder and Petryshyn [

Let

If

If

In [

In [

It is well known that the quasi-variational inclusion problem in the setting of Hilbert spaces has been extensively studied in the literature; see, for instance, [

Motivated and inspired by Zhang et al. [

Throughout this paper, we denote by

Let

A Banach space

The norm of a Banach space

Let

A Banach space

A Banach space

A Banach space

A mapping

nonexpansive if

accretive if for all

The conception of strict pseudocontractions was firstly introduced by Browder and Petryshyn [

The class of strictly pseudocontractive mappings has been studied by several authors (see, e.g., [

If

A set-valued mapping

accretive if for any

Let

In order to prove our main results, we need the following lemmas.

Let

Let

If

where

If

Let

Let

Let

Let

Let

Indeed, for all

If

Let

Assume that

From the definition of

Lemma

Assume that

Given

Given

Let

Suppose that

Let

Next we give a weak convergence theorem in a Banach space

Let

We divide the proof into several steps.

Putting

Let

Noticing (

Since

Indeed, it suffices to show

Compared with the known results in the literature, our results are very different from those in the following aspects.

Theorem

Theorem

In the following, we give a strong convergence theorem in a Banach space

Let

Let

It follows from Lemmas

Hence, to show the desired result, it suffices to prove that

Again from Lemmas

From (

By Lemmas

Theorem

from the problem of finding an element of

from a fixed element

Theorem

from Hilbert spaces to uniformly convex and

from finding a common element of the set of solutions to the variational inclusion problem and the set of fixed points of nonexpansive mappings to finding a common element of the set of solutions to the variational inclusion problem and the set of fixed points of

from a fixed element

from a fixed positive number

As a direct consequence of Theorem

Let

Using Corollary

for all

for all

Then, the mathematical model related to equilibrium problems (with respect to

The following lemma appears implicitly in Blum and Oettli [

Let

The following lemma was also given in Combettes and Hirstoaga [

Assume that

We call such

Let

Let

for all

for all

Put

The authors declare that they have no competing interests.

The work of L. C. Ceng was partially supported by the National Science Foundation of China (11071169), Ph.D. Program Foundation of Ministry of Education of China (20123127110002).