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

*Fixed Point Theory and Applications*
**volume 2011**, Article number: 368137 (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.

## 1. Introduction and Preliminaries

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

Let be a nonempty closed convex subset of and a nonlinear mapping. In this paper, we use to denote the fixed point set of . Recall that the mapping is said to be nonexpansive if

is said to be -strictly pseudocontractive if there exists a constant such that

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.

Let be a mapping. Recall that is said to be monotone if

is said to be inverse strongly monotone if there exists a constant such that

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 .

Recall that is said to be monotone if

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 .

Recall that the classical variational inequality problem is to find such that

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.

Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of into , and let be a nonexpansive mapping of into itself such that . Suppose that and is given by

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

then converges strongly to .

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

Theorem YY.

Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of into , and let be a nonexpansive mapping of into itself such that , where denotes the set of solutions of a variational inequality for the -inverse-strongly monotone mapping. Suppose that and are given by

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]).

Let be a nonempty closed convex subset of a Hilbert space a mapping, and a maximal monotone mapping. Then,

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]).

Let and be bounded sequences in a Hilbert space , and let be a sequence in with

Suppose that for all integers and

Then, .

Lemma 1.5 (see [29]).

Assume that is a sequence of nonnegative real numbers such that

where is a sequence in and is a sequence such that

(a),

(b) or .

Then, .

Lemma 1.6 (see [24]).

Let be a Hilbert space and a maximal monotone operator on . Then, the following holds:

where and .

## 2. Main Results

Theorem 2.1.

Let be a real Hilbert space and a nonempty close and convex subset of . Let and two maximal monotone operators such that and , respectively. Let be a -strict pseudocontraction, an -inverse strongly monotone mapping, and a -inverse strongly monotone mapping. Assume that . Let be a sequence generated in the following manner:

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.

Note that and are nonexpansive for each fixed . Indeed, we see from the restriction (a) that

This shows that is nonexpansive for each fixed , so is . Put

In view of the restriction (c), we obtain from Lemma 1.2 that is a nonexpansive mapping with for each fixed . Fixing and since and are nonexpansive, we see that

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

Step 2.

Show that as .

Notice from Lemma 1.6 that

where is an appropriate constant such that

Put

In a similar way, we can obtain from Lemma 1.6 that

where is an appropriate constant such that

Substituting (2.5) into (2.8) yields that

where is an appropriate constant such that

It follows from (2.10) that

where is an appropriate constant such that

Put

Note that

It follows from (2.12) that

This in turn implies from the restrictions (a)–(e) that

From Lemma 1.4, we obtain that

Notice that

It follows that

This completes Step 2.

Step 3.

Show that as .

Since and are nonexpansive, we see that

It follows from (2.21) that

This in turn implies that

In view of (2.20), we see from the restrictions (a), (d), and (e) that

It follows from (2.22) that

This in turn implies that

In view of (2.20), we see from the restrictions (a), (d), and (e) that

Since is firmly nonexpansive, we obtain that

This in turn implies that

In a similar way, we can obtain that

In view of (2.30), we see that

It follows that

In view of (2.25), we obtain from the restrictions (d) and (e) that

Notice from (2.31), we see that

It follows that

In view of (2.28), we obtain from the restrictions (d) and (e) that

Combining (2.34) with (2.37) yields that

Note that

In view of (2.20), we see from the restriction (d) that

Note that

From (2.38) and (2.40), we get from the restriction (c) that

Notice that

In view of (2.38) and (2.42), we see from Lemma 1.3 that

This completes Step 3.

Step 4.

Show that , where .

To show it, we may choose a subsequence of such that

Since is bounded, we can choose a subsequence of converging weakly to . We may, without loss of generality, assume that , where denotes the weak convergence. Next, we prove that . In view of (2.44), we can conclude from Lemma 1.3 that easily. Notice that

Let . Since is monotone, we have

In view of the restriction (a), we see from (2.34) that

This implies that , that is, . In similar way, we can obtain that . This proves that . It follows from (2.45) that

This completes Step 4.

Step 5.

Show that as .

Notice that

This in turn implies that

In view of (2.49), we conclude from Lemma 1.5 that

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.

Let be a real Hilbert space and a nonempty close and convex subset of . Let and be two maximal monotone operators such that and , respectively. Let be a nonexpansive mapping, an -inverse strongly monotone mapping and a -inverse strongly monotone mapping. Assume that . Let be a sequence generated in the following manner:

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.

Let be a nonempty closed convex subset of a real Hilbert space and the metric projection from onto . Let be a -strict pseudocontraction, an -strict pseudocontraction, and a -strict pseudocontraction. Assume that . Let be a sequence generated in the following manner:

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.

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## 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).

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Yu, L., Liang, M. Convergence of Iterative Sequences for Fixed Point and Variational Inclusion Problems.
*Fixed Point Theory Appl* **2011, **368137 (2011). https://doi.org/10.1155/2011/368137

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### Keywords

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