- 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 .
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,
Then is a strict pseudo-contraction but not a strong pseudo-contraction.
Then, is a strong pseudo-contraction but not a strict pseudo-contraction.
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.
Theorem 1 ST.
Then converges strongly to the element of nearest to
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 , 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.
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:
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  if, for each sequence in which converges weakly to point ,
That is, is strongly monotone with the coefficient .
This completes the proof.
First, we give a convergence theorem for a non-expansive semigroup by Moudafi's viscosity approximation methods with -contractions.
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.
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.
From Theorem 2.3, we see that the composite mapping has a unique fixed point in .
- 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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