Composite Implicit General Iterative Process for a Nonexpansive Semigroup in Hilbert Space

Abstract

Let be nonempty closed convex subset of real Hilbert space . Consider a nonexpansive semigroup with a common fixed point, a contraction with coefficient , and a strongly positive linear bounded operator with coefficient . Let . It is proved that the sequence generated iteratively by converges strongly to a common fixed point which solves the variational inequality for all .

1. Introduction and Preliminaries

Let be a closed convex subset of a Hilbert space , recall that is nonexpansive if for all . Denote by the set of fixed points of , that is, .

Recall that a family of mappings from into itself is called a nonexpansive semigroup on if it satisfies the following conditions:

• (i) for all ;

• (ii) for all ;

• (iii) for all and ;

• (iv) for all is continuous.

We denote by the set of all common fixed points of , that is, . It is known that is closed and convex.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems (see, e.g., [1–5] and the references therein). A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :

(1.1)

where is the fixed point set of a nonexpansive mapping on , and is a given point in . Assume that is strongly positive, that is, there is a constant with the property

(1.2)

It is well known that is closed convex (cf. [6]). In[3] (see also [4]), it is proved that the sequence defined by the iterative method below, with the initial guess chosen arbitrarily,

(1.3)

converges strongly to the unique solution of the minimization problem (1.1) provided that the sequence satisfies certain conditions.

On the other hand, Moudafi [7] introduced the viscosity approximation method for nonexpansive mappings (see [8] for further developments in both Hilbert and Banach spaces). Let be a contraction on . Starting with an arbitrary initial , define a sequence recursively by

(1.4)

where is a sequence in (). It is proved [7, 8] that under certain appropriate conditions imposed on , the sequence generated by (1.4) strongly converges to the unique solution in of the variational inequality

(1.5)

Recently, Marino and Xu [9] combined the iterative method (1.3) with the viscosity approximation method (1.4) considering the following general iteration process:

(1.6)

and proved that if the sequence satisfies appropriate conditions, then the sequence generated by (1.6) converges strongly to the unique solution of the variational inequality

(1.7)

which is the optimality condition for the minimization problem

(1.8)

where is a potential function for (i.e., , for ).

In this paper, motivated and inspired by the idea of Marino and Xu [9], we introduce the composite implicit general iteration process (1.9) as follows:

(1.9)

where , and investigate the problem of approximating common fixed point of nonexpansive semigroup which solves some variational inequality. The results presented in this paper extend and improve the main results in Marino and Xu [9], and the methods of proof given in this paper are also quite different.

In what follows, we will make use of the following lemmas. Some of them are known; others are not hard to derive.

Lemma 1.1 (Marino and Xu [9]).

Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma (Shimizu and Takashi [10]).

Let be a nonempty bounded closed convex subset of and let be a nonexpansive semigroup on , then for any ,

(1.10)

Lemma 1.3.

Let be a nonempty bounded closed convex subset of a Hilbert space and let be a nonexpansive semigroup on . If is a sequence in satisfying the following properties:

• (i)
• (ii)

where denote that converges weakly to , then .

Proof.

This lemma is the continuous version of Lemma 2.3 of Tan and Xu [11]. This proof given in [11] is easily extended to the continuous case.

2. Main Results

Lemma 2.1.

Let be a Hilbert space, a closed convex subset of , let be a nonexpansive semigroup on , is a sequence, then is monotone.

Proof.

In fact, for all ,

(2.1)

Theorem 2.2.

Let be nonempty closed convex subset of real Hilbert space , suppose that is a fixed contractive mapping with coefficient , and is a nonexpansive semigroup on such that is nonempty, and is a strongly positive linear bounded operator with coefficient , , are real sequences such that

(2.2)

then for any , there is a unique such that

(2.3)

and the iteration process converges strongly to the unique solution of the variational inequality for all .

Proof.

Our proof is divided into five steps.

Since , as , we may assume, with no loss of generality, that , for all .

(i) is bounded.

Firstly, we will show that the mapping defined by

(2.4)

is a contraction. Indeed, from Lemma 1.1, we have for any that

(2.5)

Let be the unique fixed point of . Thus,

(2.6)

is well defined. Next, we will show that is bounded.

Pick any to obtain

(2.7)

(2.8)

Also

(2.9)

Substituting (2.9) into (2.8), we obtain that

(2.10)

Thus is bounded.

(ii).

Denote that , since is bounded, and , are also bounded, From (2.6) and , we have

(2.11)

Let , then is a nonempty bounded closed convex subset of and -invariant. Since and is bounded, there exists such that , it follows from Lemma 1.2 that

(2.12)

From (2.11) and (2.12), we have

(2.13)

(iii)There exists a subsequence of such that and is the unique solution of the following variational inequality:

(2.14)

Firstly since

(2.15)

From condition and the boundedness of , we obtain that . Again by boundedness of , we know that there exists a subsequence of such that . Then . From Lemma 1.3 and step (ii), we have that .

Next we will prove that solves the variational inequality (2.14). Since

(2.16)

we derive that

(2.17)

It follows that, for all ,

(2.18)

Using Lemma 2.1, we have from (2.18) that

(2.19)

Now replacing in (2.19) with and letting , we notice that

(2.20)

and from condition and boundedness of , we have

(2.21)

For , we obtain

(2.22)

From [9, Theorem 3.2], we know that the solution of the variational inequality (2.14) is unique. That is, is a unique solution of (2.14).

• (iv)
  

  (2.23)

where is obtained in step (iii).

To see this, there exists a subsequence of such that

(2.24)

we may also assume that , then , note from step (ii) that in virtue of Lemma 1.2. It follows from the variational inequality (2.14) that

(2.25)

So (2.23) holds thank to (2.14).

(v).

Finally, we will prove . Since

(2.26)

Next, we calculate

(2.27)

Thus it follows from (2.26) that

(2.28)

Thus

(2.29)

Since is bounded, we can take a constant such that

(2.30)

for all . It then follows from (2.29) that

(2.31)

Then

(2.32)

From condition , and (2.23), we conclude that

(2.33)

So . This completes the proof of the Theorem 2.2.

It follows from the above proof that Theorem 2.2 is valid for nonexpansive mappings. Thus, we have that Corollaries 2.3 and 2.4 are two special cases of Theorem 2.2.

Corollary 2.3.

Let be a nonexpansive mapping from nonempty closed convex subset of a Hilbert space to , is generated by the following algorithm:

(2.36)

where are real sequences such that

(2.35)

then for any , the sequence above converges strongly to the unique solution of the variational inequality for all .

Corollary 2.4.

Let be a nonexpansive mapping from nonempty closed convex subset of a Hilbert space to , is generated by the following algorithm:

(2.34)

where is a sequence in () satisfying the following condition: , then the sequence converges strongly to the unique solution of the variational inequality for all .