Moudafi's Viscosity Approximations with Demi-Continuous and Strong Pseudo-Contractions for Non-Expansive Semigroups

Abstract

We consider viscosity approximation methods with demi-continuous strong pseudo-contractions for a non-expansive semigroup. Strong convergence theorems of the purposed iterative process are established in the framework of Hilbert spaces.

1. Introduction and Preliminaries

Throughout this paper, we assume that is a real Hilbert space and denote by the set of nonnegative real numbers. Let be a nonempty closed and convex subset of and a nonlinear mapping. We use to denote the fixed point set of .

Recall that the mapping is said to be an -contraction if there exists a constant such that

(1.1)

is said to be non-expansive if

(1.2)

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

(1.3)

Note that the class of strict pseudo-contractions strictly includes the class of non-expansive mappings as a special case. That is, is non-expansive if and only if the coefficient . It is also said to be pseudo-contractive if . That is,

(1.4)

is said to be strongly pseudo-contractive if there exists a positive constant such that is pseudo-contractive. Clearly, the class of strict pseudo-contractions falls into the one between classes of non-expansive mappings and pseudo-contractions. We remark also that the class of strongly pseudo-contractive mappings is independent of the class of strict pseudo-contractions (see, e.g., [1, 2]).

The following examples are due to Chidume and Mutangadura [3] and Zhou [4].

Example 1.1.

Take and define by

(1.5)

Then is a strict pseudo-contraction but not a strong pseudo-contraction.

Example 1.2.

Take and define by

(1.6)

Then, is a strong pseudo-contraction but not a strict pseudo-contraction.

Example 1.3.

Take and , , . If , we define to be Define by

(1.7)

Then, is a Lipschitz pseudo-contraction but not a strict pseudo-contraction.

It is very clear that, in a real Hilbert space , (1.4) is equivalent to

(1.8)

is strongly pseudo-contractive if and only if there exists a positive constant such that

(1.9)

for all .

Let be a strongly continuous semigroups of non-expansive mappings on a closed convex subset of a Hilbert space , that is,

(a)for each , is a non-expansive mapping on

(b) for all

(c) for all

(d)for each , the mapping from into is continuous.

We denote by the set of common fixed points of , that is,

(1.10)

We know that is nonempty if is bounded (see [5]). In [6], Shioji and Takahashi proved the following theorem:

Theorem 1 ST.

Let be a closed convex subset of a Hilbert space . Let be a strongly continuous semigroup of non-expansive mappings on such that . Let and be sequences of real numbers satisfying , , and . Fix  and define a sequence in by

(1.11)

Then converges strongly to the element of nearest to

Suzuki [9] improved the results of Shioji and Takahashi [6] and proved the following theorem:

Theorem 1 S.

Let be a closed convex subset of a Hilbert space . Let be a strongly continuous semigroup of non-expansive mappings on such that . Let and be sequences of real numbers satisfying , and . and define a sequence in by

(1.12)

Then converges strongly to the element of nearest to

Recently, The so-called viscosity approximation methods have been studied by many author. They are very important because they are applied to convex optimization, linear programming, monotone inclusions and elliptic differential equations. In [8], Moudafi proposed the viscosity approximation method of selecting a particular fixed point of a given non-expansive mapping in Hilbert spaces. If is a Hilbert space, is a non-expansive self-mapping on a nonempty closed convex of and is a contraction, he proved the following result.

Theorem 1 M.

The sequence generated by the scheme

(1.13)

converges strongly to the unique solution of the variational inequality:

(1.14)

where is a sequence of positive numbers tending to zero.

In this paper, motivated by Moudafi [8], Shioji and Takahashi [6], Suzuki [9] and Xu [10], we introduce the following implicit iterative scheme:

(1.15)

where is a demi-continuous and strong pseudo-contraction, and prove that the sequence generated by the above iterative process converge strongly to a common fixed point . Also, we show that the point solves the variational inequality

(1.16)

Our results mainly improve and extend the corresponding results announced by Moudafi [8], Shioji and Takahashi [6], Suzuki [9], Xu [10] and some others.

In order to prove our main result, we need the following lemmas and definitions:

Let be linear normed spaces. is said to be demi-continuous if, for any we have as , where and denote strong and weak convergence, respectively.

Recall that a space satisfies Opial's condition [11] if, for each sequence in which converges weakly to point ,

(1.17)

Lemma 1.4.

Let be a closed convex subset of a Hilbert space , be a strong pseudo-contraction with the coefficient . Then

(1.18)

That is, is strongly monotone with the coefficient .

Proof.

From the definition of strongly pseudo-contractions, one sees that

(1.19)

Therefore, we have

(1.20)

This completes the proof.

Remark 1.5.

If is non-expansive, then

(1.21)

2. Main Results

First, we give a convergence theorem for a non-expansive semigroup by Moudafi's viscosity approximation methods with -contractions.

Theorem 2.1.

Let be a nonempty closed and convex subset of a Hilbert space . Let be a strongly continuous semigroup of non-expansive mappings from into itself such that . Let be an -contraction. Let and be sequences of real numbers satisfying , and . Define a sequence in the following manner:

(2.1)

Then converges strongly to which solves the following variational inequality:

(2.2)

Proof.

Define a sequence in the following manner

(2.3)

From Theorem S, one sees that

(2.4)

Therefore, it is sufficient to prove that

(2.5)

Noticing that

(2.6)

one has

(2.7)

It follows that

(2.8)

From (2.4), one obtains that (2.5) holds. This completes the proof.

Remark 2.2.

If , a fixed point, for all , then Theorem 2.1 is reduced to Suzuki's results [9]. Theorem 2.1 also can be viewed as an improvement of the corresponding results in Shioji and Takahashi [6].

The class of pseudo-contractions is one of the most important classes of mappings among nonlinear mappings. Browder [1] proved the first existence result of fixed point for demi-continuous pseudo-contractions in the framework of Hilbert space. During the past 40 years or so, mathematicians have been devoting to the studies on the existence and convergence of fixed points of nonexpansive mappings and pseudo-contractive mappings. See, for example, [1–26].

Assume that is a metric projection from a Hilbert space to its nonempty closed convex subset and is a -contraction. It is easy to see that the mapping has a unique fixed point in . That is why Theorem 2.1 can be deduced from Theorem S easily. What happens if we relax from -contraction to strong pseudo-contraction? Does Theorem 2.1 still holds if is a strong pseudo-contraction? Since we don't know whether the mapping , where is a strong pseudo-contraction, has a unique fixed point or not, we can not get the desired results from Theorem S.

Next, we give the second convergence theorem for the non-expansive semigroup by Moudafi's viscosity approximation methods with strong pseudo-contractions.

Theorem 2.3.

Let be a nonempty closed and convex subset of a Hilbert space . Let be a strongly continuous semigroup of non-expansive mappings from into itself such that . Let be bounded, demi-continuous and strong pseudo-contraction with the coefficient . Let and be sequences of real numbers satisfying , and . Define a sequence by the following manner:

(2.9)

Then converges strongly to which solves the following variational inequality:

(2.10)

Proof.

First, we show that the fixed point equation (2.10) is well-defined. For any , define a mapping as follows

(2.11)

For any one has

(2.12)

This shows is demi-continuous and strong pseudo-contraction with the efficient . From Lan and Wu [17, Theorem ], one sees that has a unique fixed point, denoted , which uniquely solves the fixed point equation

(2.13)

This is, (2.9) is well-defined. The uniqueness of the solution of the variational inequality (2.10) is a consequence of the strong monotonicity of . Suppose both are solutions to (2.10). It follows that

(2.14)

Adding up (2.14), one obtains

(2.15)

Since Lemma 1.4, one sees that . Next, we use to denote the unique solution of the variational inequality (2.10).

Next, we show that is bounded. Indeed, for any , we have

(2.16)

from which it follows that

(2.17)

That is,

(2.18)

This implies that is bounded. Let be an arbitrary subsequence of Then there exists a subsequence of which converges weakly to a point .

Next, we show that In fact, put , and for all . Fix Noticing that

(2.19)

we have

(2.20)

From Opial's condition, we have Therefore, In the inequality (2.17), replacing with , we have

(2.21)

Taking the limit as in (2.21), we obtain

(2.22)

Since the subsequence is arbitrary, it follows that converges strongly to

Finally, we prove that is a solution of the variational inequality (2.10). From (2.9), one sees

(2.23)

For any , it follows from (1.21) that

(2.24)

Letting , one sees

(2.25)

That is, is the unique solution to the variational inequality (2.10). This completes the proof.

Remark 2.4.

From Theorem 2.3, we see that the composite mapping has a unique fixed point in .