- Research Article
- Open Access
Moudafi's Viscosity Approximations with Demi-Continuous and Strong Pseudo-Contractions for Non-Expansive Semigroups
© Changqun Wu et al. 2010
- Received: 6 October 2009
- Accepted: 10 December 2009
- Published: 12 January 2010
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.
- Hilbert Space
- Variational Inequality
- Convex Subset
- Nonexpansive Mapping
- Real Hilbert Space
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 .
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,
Theorem 1 ST.
Theorem 1 S.
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 , 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 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
In order to prove our main result, we need the following lemmas and definitions:
Recall that a space satisfies Opial's condition  if, for each sequence in which converges weakly to point ,
This completes the proof.
From (2.4), one obtains that (2.5) holds. This completes the proof.
The class of pseudo-contractions is one of the most important classes of mappings among nonlinear mappings. Browder  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.
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