- Research Article
- Open Access

# Convergence of Iterative Sequences for Fixed Point and Variational Inclusion Problems

- Li Yu
^{1}Email author and - Ma Liang
^{1}

**2011**:368137

https://doi.org/10.1155/2011/368137

© Li Yu and Ma Liang. 2011

**Received:**14 November 2010**Accepted:**8 February 2011**Published:**1 March 2011

## Abstract

An iterative process is considered for finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and an inverse strongly monotone mapping. Strong convergence theorems of common elements are established in real Hilbert spaces.

## Keywords

- Variational Inequality
- Monotone Mapping
- Nonexpansive Mapping
- Common Element
- Maximal Monotone

## 1. Introduction and Preliminaries

Throughout this paper, we always assume that is a real Hilbert space with the inner product and the norm .

The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn [1] in 1967. It is easy to see that every nonexpansive mapping is a 0-strictly pseudocontractive mapping.

For such a case, is also said to be -inverse strongly monotone.

Let be a set-valued mapping. The set defined by is said to be the domain of . The set defined by is said to be the range of . The set defined by is said to be the graph of .

is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently, is maximal monotone if for all . For a maximal monotone operator on and , we may define the single-valued resolvent . It is known that is firmly nonexpansive and .

Denote by of the solution set of (1.6). It is known that is a solution to (1.6) if and only if is a fixed point of the mapping , where is a constant and is the identity mapping.

Recently, many authors considered the convergence of iterative sequences for the variational inequality (1.6) and fixed point problems of nonlinear mappings see, for example, [1–32].

In 2005, Iiduka and Takahashi [7] proved the following theorem.

Theorem IT.

then converges strongly to .

In 2007, Y. Yao and J.-C. Yao [31] further obtained the following theorem.

Theorem YY.

where , and are three sequences in and is a sequence in . If , , , and are chosen so that for some with and

(a) , for all ,

(b) ,

(c) ,

(d) ,

then converges strongly to .

In this work, motivated by the above results, we consider the problem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and a inverse strongly monotone mapping. Strong convergence theorems of common elements are established in real Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi [7] and Y. Yao and J.-C. Yao [31].

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

Lemma 1.1 (see [22]).

Lemma 1.2 (see [1]).

Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudocontraction with a fixed point. Define by for each . If , then is nonexpansive with .

Lemma 1.3 (see [25]).

Let be a nonempty closed convex subset of a Hilbert space and a -strict pseudocontraction. Then,

(a) is -Lipschitz,

(b) is demi-closed, this is, if is a sequence in with and , then .

Lemma 1.4 (see [28]).

Then, .

Lemma 1.5 (see [29]).

where is a sequence in and is a sequence such that

(a) ,

(b) or .

Then, .

Lemma 1.6 (see [24]).

where and .

## 2. Main Results

Theorem 2.1.

where is a fixed element, and , is a sequence in , is a sequence in and , , , and are sequences in . Assume that the following restrictions are satisfied:

(a) , ,

(b) , ,

(c) , ,

(d) , ,

(e) .

Then, the sequence converges strongly to .

Proof.

The proof is split into five steps.

Step 1.

Show that is bounded.

By mathematical inductions, we see that is bounded and so is . This completes Step 1.

Step 2.

Show that as .

This completes Step 2.

Step 3.

Show that as .

This completes Step 3.

Step 4.

Show that , where .

This completes Step 4.

Step 5.

Show that as .

This completes Step 5. This whole proof is completed.

If is a nonexpansive mapping and , then Theorem 2.1 is reduced to the following.

Corollary 2.2.

where is a fixed element, and , is a sequence in , is a sequence in and , and are sequences in . Assume that the following restrictions are satisfied:

(a) ,

(b) ,

(c) ,

(d) .

Then, the sequence converges strongly to .

Next, we consider the problem of finding common fixed points of three strict pseudocontractions.

Theorem 2.3.

where is a fixed element, is a sequence in , is a sequence in , and , , , and are sequences in . Assume that the following restrictions are satisfied

(a) ,

(b) ,

(c) ,

(d) ,

(e) .

Then, the sequence converges strongly to .

Proof.

Putting , we see that is -inverse strongly monotone. We also have and . Putting , we see that is -inverse strongly monotone. We also have and . In view of Theorem 2.1, we can obtain the desired results immediately.

## Declarations

### Acknowledgments

The authors are extremely grateful to the referees for useful suggestions that improved the contents of the paper. This work was supported by the National Natural Science Foundation of China under Grant no. 70871081 and Important Science and Technology Research Project of Henan province, China (102102210022).

## Authors’ Affiliations

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