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In this paper we consider a problem that consists of finding a zero to the sum of two monotone operators. One method for solving such a problem is the forward-backward splitting method. We present some new conditions that guarantee the weak convergence of the forward-backward method. Applications of these results, including variational inequalities and gradient projection algorithms, are also considered.

It is well known that monotone inclusions problems play an important role in the theory of nonlinear analysis. This problem consists of finding a zero of maximal monotone operators. However, in some examples such as convex programming and variational inequality problems, the operator is needed to be decomposed of the sum of two monotone operators (see, e.g., [

On the other hand, we observe that problem (

The rest of this paper is organized as follows. In Section

Throughout the paper,

Let

The following assertions hold.

for all

Assume that

The following lemma is known as the demiclosedness principle for nonexpansive mappings.

Let

A multivalued operator

In what follows, we shall assume that

For

For

Assume that

Let

the sequence

if

In [

Suppose the following conditions are satisfied:

We first show that

Next let us show

We can also present another condition for the weak convergence of (

Suppose that the following conditions are satisfied:

Compared with the proof of Theorem

Applying Theorem

Suppose that the following conditions are satisfied:

Corollary

Let

Let

Suppose the following conditions are satisfied:

Suppose that the following conditions are satisfied:

Consider the optimization problem of finding a point

Assume that

Assume that

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to express their sincere thanks to the anonymous referees and editors for their careful review of the paper and the valuable comments, which have greatly improved the earlier version of this paper. This work is supported by the National Natural Science Foundation of China (Grant nos. 11301253 and 11271112), the Basic and Frontier Project of Henan (no. 122300410268), and the Science and Technology Key Project of Education Department of Henan Province (14A110024).