Convergence of Iterative Sequences for Fixed Point and Variational Inclusion Problems

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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

(1.8)

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

(1.9)

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,

(1.10)

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

(1.11)

Suppose that for all integers and

(1.12)

Then, .

Lemma 1.5 (see [29]).

Assume that is a sequence of nonnegative real numbers such that

(1.13)

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:

(1.14)

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:

(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.

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

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

where is an appropriate constant such that

(2.6)

Put

(2.7)

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

(2.8)

where is an appropriate constant such that

(2.9)

Substituting (2.5) into (2.8) yields that

(2.10)

where is an appropriate constant such that

(2.11)

It follows from (2.10) that

(2.12)

where is an appropriate constant such that

(2.13)

Put

(2.14)

Note that

(2.15)

It follows from (2.12) that

(2.16)

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

(2.17)

From Lemma 1.4, we obtain that

(2.18)

Notice that

(2.19)

It follows that

(2.20)

This completes Step 2.

Step 3.

Show that as .

Since and are nonexpansive, we see that

(2.21)

(2.22)

It follows from (2.21) that

(2.23)

This in turn implies that

(2.24)

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

(2.25)

It follows from (2.22) that

(2.26)

This in turn implies that

(2.27)

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

(2.28)

Since is firmly nonexpansive, we obtain that

(2.29)

This in turn implies that

(2.30)

In a similar way, we can obtain that

(2.31)

In view of (2.30), we see that

(2.32)

It follows that

(2.33)

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

(2.34)

Notice from (2.31), we see that

(2.35)

It follows that

(2.36)

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

(2.37)

Combining (2.34) with (2.37) yields that

(2.38)

Note that

(2.39)

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

(2.40)

Note that

(2.41)

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

(2.42)

Notice that

(2.43)

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

(2.44)

This completes Step 3.

Step 4.

Show that , where .

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

(2.45)

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

(2.46)

Let . Since is monotone, we have

(2.47)

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

(2.48)

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

(2.49)

This completes Step 4.

Step 5.

Show that as .

Notice that

(2.50)

This in turn implies that

(2.51)

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

(2.52)

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:

(2.53)

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:

(2.54)

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.