- Research Article
- Open Access

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

- Changqun Wu
^{1}Email author, - SunYoung Cho
^{2}and - Meijuan Shang
^{3}

**2010**:645498

https://doi.org/10.1155/2010/645498

© Changqun Wu et al. 2010

**Received:**6 October 2009**Accepted:**10 December 2009**Published:**12 January 2010

## 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.

## Keywords

- Hilbert Space
- Variational Inequality
- Convex Subset
- Nonexpansive Mapping
- Real Hilbert Space

## 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

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,

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

Example 1.1.

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

Example 1.2.

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

Example 1.3.

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

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

Theorem 1 ST.

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.

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.

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:

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

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 ,

Lemma 1.4.

That is, is strongly monotone with the coefficient .

Proof.

This completes the proof.

Remark 1.5.

## 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.

Proof.

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.

Proof.

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

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

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

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 .

## Authors’ Affiliations

## References

- Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space.
*Proceedings of the National Academy of Sciences of the United States of America*1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar - Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space.
*Journal of Mathematical Analysis and Applications*1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Chidume CE, Mutangadura SA: An example of the Mann iteration method for Lipschitz pseudocontractions.
*Proceedings of the American Mathematical Society*2001, 129(8):2359–2363. 10.1090/S0002-9939-01-06009-9MathSciNetView ArticleMATHGoogle Scholar - Zhou H: Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces.
*Journal of Mathematical Analysis and Applications*2008, 343(1):546–556. 10.1016/j.jmaa.2008.01.045MathSciNetView ArticleMATHGoogle Scholar - Browder FE: Nonexpansive nonlinear operators in a Banach space.
*Proceedings of the National Academy of Sciences of the United States of America*1965, 54: 1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleMATHGoogle Scholar - Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces.
*Nonlinear Analysis: Theory, Methods & Applications*1998, 34(1):87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho YJ, Kang SM, Zhou H: Convergence theorems of common fixed points for a family of Lipschitz quasi-pseudocontractions.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 71(1–2):685–690. 10.1016/j.na.2008.10.102MathSciNetView ArticleMATHGoogle Scholar - Moudafi A: Viscosity approximation methods for fixed-points problems.
*Journal of Mathematical Analysis and Applications*2000, 241(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar - Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces.
*Proceedings of the American Mathematical Society*2003, 131(7):2133–2136. 10.1090/S0002-9939-02-06844-2MathSciNetView ArticleMATHGoogle Scholar - Xu H-K: Viscosity approximation methods for nonexpansive mappings.
*Journal of Mathematical Analysis and Applications*2004, 298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings.
*Bulletin of the American Mathematical Society*1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Browder FE: Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces.
*Archive for Rational Mechanics and Analysis*1967, 24: 82–90.MathSciNetView ArticleMATHGoogle Scholar - Bruck RE Jr.: A strongly convergent iterative solution of for a maximal monotone operator in Hilbert space.
*Journal of Mathematical Analysis and Applications*1974, 48: 114–126. 10.1016/0022-247X(74)90219-4MathSciNetView ArticleMATHGoogle Scholar - Chidume CE, Zegeye H: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps.
*Proceedings of the American Mathematical Society*2004, 132(3):831–840. 10.1090/S0002-9939-03-07101-6MathSciNetView ArticleMATHGoogle Scholar - Cho YJ, Qin X: Viscosity approximation methods for a family of -accretive mappings in reflexive Banach spaces.
*Positivity*2008, 12(3):483–494. 10.1007/s11117-007-2181-8MathSciNetView ArticleMATHGoogle Scholar - Deimling K: Zeros of accretive operators.
*Manuscripta Mathematica*1974, 13: 365–374. 10.1007/BF01171148MathSciNetView ArticleMATHGoogle Scholar - Lan KQ, Wu JH: Convergence of approximants for demicontinuous pseudo-contractive maps in Hilbert spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2002, 49(6):737–746. 10.1016/S0362-546X(01)00130-4MathSciNetView ArticleMATHGoogle Scholar - Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces.
*Journal of Mathematical Analysis and Applications*2007, 329(1):415–424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleMATHGoogle Scholar - Qin X, Su Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 67(6):1958–1965. 10.1016/j.na.2006.08.021MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho YJ, Kang SM, Shang M: A hybrid iterative scheme for asymptotically
*k*-strict pseudo-contractions in Hilbert spaces.*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(5):1902–1911. 10.1016/j.na.2008.02.090MathSciNetView ArticleMATHGoogle Scholar - Qin X, Kang SM, Shang M: Strong convergence theorems for accretive operators in Banach spaces.
*Fixed Point Theory*2008, 9(1):243–258.MathSciNetMATHGoogle Scholar - Qin X, Cho YJ, Kang JI, Kang SM: Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces.
*Journal of Computational and Applied Mathematics*2009, 230(1):121–127. 10.1016/j.cam.2008.10.058MathSciNetView ArticleMATHGoogle Scholar - Rhoades BE: Fixed point iterations using infinite matrices.
*Transactions of the American Mathematical Society*1974, 196: 161–176.MathSciNetView ArticleMATHGoogle Scholar - Zhou H: Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(10):2977–2983. 10.1016/j.na.2007.02.041MathSciNetView ArticleMATHGoogle Scholar - Zhou H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(11):4039–4046. 10.1016/j.na.2008.08.012MathSciNetView ArticleMATHGoogle Scholar - Zhou H: Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(9):3140–3145. 10.1016/j.na.2008.04.017MathSciNetView ArticleMATHGoogle Scholar

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