# Degree of Convergence of Iterative Algorithms for Boundedly Lipschitzian Strong Pseudocontractions

- Songnian He
^{1, 2}Email author, - Yongfu Su
^{3}and - Ronghua Li
^{1}

**2010**:210340

https://doi.org/10.1155/2010/210340

© Songnian He et al. 2010

**Received: **25 September 2010

**Accepted: **19 December 2010

**Published: **29 December 2010

## Abstract

Let be a nonempty closed convex subset of a real Hilbert space and let be a boundedly Lipschitzian strong pseudo-contraction with a nonempty fixed point set. Three iterative algorithms are proposed for approximating the unique fixed point of ; one of them is for the self-mapping case, and the others are for the nonself-mapping case. Not only the strong convergence, but also the degree of convergence of the three iterative algorithms is obtained. Some numerical results corresponding to the self-mapping case are given which show advantages of our methods. As an application of our results, adopting the regularization idea, we also propose implicit and explicit algorithms for approximating a fixed point of a boundedly Lipschitzian pseudocontractive self-mapping from into itself, respectively.

## 1. Introduction and Preliminaries

for all . In this case, we also call a -strong pseudocontraction. Using (1.2), it is easy to see that every strong pseudocontraction has at most one fixed point.

We will denote by the set of fixed points of , that is, . Let be a sequence and a point in . Then we use and to denote strong and weak convergence to of the sequence , respectively.

Among classes of nonlinear mappings, the class of pseudocontractions is one of the most important classes of mappings. This is mainly due to the fact that there is a precise corresponding relation between the class of pseudocontractions and the class of monotone mappings. A mapping is monotone (i.e., for all ) if and only if is pseudo-contractive, where and denotes the identity mapping on .

Within the past 40 years or so, mathematicians have been devoting their study to the existence and iterative construction of fixed points for pseudocontractions and of zeros for monotone mappings (see, e.g., [1–18]). However, most of these algorithms have no estimation of degree of convergence even if for strong pseudocontractions in setting of Hilbert spaces. Everyone knows that it is very important to get the degree of convergence for an algorithm in computing science.

The main purpose of this paper is to consider the iterative algorithms for approximating the unique fixed point (if the set of fixed points is not empty) of a boundedly Lipschitzian strong pseudocontraction defined on a nonempty closed convex subset of a real Hilbert space. Three iterative algorithms are proposed; one of them is for the self-mapping case, and the others are for the nonself-mapping case. Not only the strong convergence, but also the degree of convergence of the three iterative algorithms is obtained. Some numerical results corresponding to the self-mapping case are given which show advantages of our methods. As an application of our results, adopting the regularization idea, we also establish implicit and explicit algorithms for approximating a fixed point of a boundedly Lipschitzian pseudo-contractive self-mapping from into itself, respectively.

In order to give our main results, let us recall a basic existence result for fixed points for continuous strong pseudocontractions which was proved by Deimling [6] in 1974.

Theorem 1.1 (Deimling [6]).

for each , where denotes the distance from the point to the subset of . Then has a unique fixed point.

Corollary 1.2 (see [6]).

Let be a closed convex subset of a real Banach space , and let be a continuous -strong pseudocontraction, then has a unique fixed point.

We also need some facts which are listed as lemmas below.

Lemma 1.3 (see [7]).

is closed and convex.

Lemma 1.4 (see, e.g., [9]).

Lemma 1.5 (see [18]).

Let be a nonempty closed convex subset of a real Hilbert space and a demicontinuous pseudo-contractive self-mapping from into itself. Then is a closed convex subset of and is demiclosed at zero.

Now we are in a position to prove main results in this paper.

## 2. Algorithms for Strongly Pseudocontractive Self-Mappings

In this section, we propose an iterative algorithm for boundedly Lipschitzian and strongly pseudo-contractive self-mappings. Since the algorithm has nothing to do with the metric projection, it is easy to realize in practical computing.

Theorem 2.1.

In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.

Proof.

Thus (2.11) together with (2.10) leads to (2.5). On the other hand, we have from (2.4), (2.5), and (2.8) that , that is, .

Thus (2.2) is obtained by using (2.5) and (2.13).

hence is a -strong pseudocontraction.

This shows that the estimation (2.2) cannot be improved.

Remark 2.2.

If , then reaches the minimum when , so is said to be optimal control parameter of process (2.1). If , then it is not difficult to verify that the optimal control parameter is zero. The same result also applies to all of the following algorithms.

If is Lipschitzian on the whole , that is, there exists a positive constant such that for all , then we obtain the following result as a special case of Theorem 2.1.

Theorem 2.3.

In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.

## 3. Algorithms for Strongly Pseudo-Contractive Nonself-Mappings

In this section, we turn to designing two iterative algorithms for boundedly Lipschitzian and strongly pseudo-contractive nonself-mappings. In this case, a boundedly Lipschitzian strong pseudocontraction may not have a fixed point, so we assume that the mapping has a unique fixed point (noting that each strong pseudocontraction has at most one fixed point). In addition, we will have to use the metric projection in the algorithms.

In fact, the first algorithm is a modification of process (2.1) as follows. We omit its proof, which is very similar to the proof of Theorem 2.1.

Theorem 3.1.

In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.

If is Lipschitzian on the whole , we have the following result as a special case of Theorem 3.1.

Theorem 3.2.

Now we give the second algorithm for the strongly pseudo-contractive nonself-mapping case.

Theorem 3.3.

Proof.

The definition of and (3.10) imply that . Using Lemma 1.3 again, the condition guarantees that is nonempty, closed, and convex. So there exists a unique element such that .

Thus the combination of (3.12) and (3.14) leads to (3.6), and converges strongly to due to the fact that .

By the same argument in the proof of Theorem 2.1, we assert that (3.6) is the optimal estimation of degree of convergence.

Remark 3.4.

The formulation of process (3.1) is simpler than that of process (3.5). But process (3.5) is believed to have faster rate of convergence than that of process (3.1) due to the fact that , in general.

## 4. Algorithms for Pseudo-Contractive Self-Mappings

Let be a nonempty closed convex subset of a real Hilbert space , and let be a boundedly Lipschitzian pseudocontraction with a nonempty fixed point set , that is, . It follows from Lemma 1.5 that is closed and convex, so the metric projection operator is well defined.

Firstly, we prove that converges strongly to a fixed point of , as . Next, we give our explicit method based on this implicit method and Theorem 2.1.

Theorem 4.1.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a boundedly Lipschitzian pseudocontraction with a nonempty fixed point set . Let , and is determined by (4.2). Then is bounded, and . Moreover, .

Proof.

Clearly, . This says that is bounded. Since is boundedly Lipschitzian, so it is not difficult to show that is also bounded. Thus we can assert that the set of weak cluster points , where .

Taking the limit as , we see that .

By Lemma 1.4, we get . This means that . Thus we have proved that .

Theorem 4.1 says that . Observe that is boundedly Lipschitzian and -strongly pseudo-contractive.

Theorem 4.2.

and the control parameter sequence such that . Then converges strongly to .

Proof.

## Declarations

### Acknowledgment

This paper is supported by Fundamental Research Funds for the Central Universities (Grant no. ZXH2009D021).

## Authors’ Affiliations

## References

- Browder FE:
**Existence of periodic solutions for nonlinear equations of evolution.***Proceedings of the National Academy of Sciences of the United States of America*1965,**53:**1100–1103. 10.1073/pnas.53.5.1100MathSciNetView 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 - 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 - Chen R, Song Y, Zhou H:
**Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings.***Journal of Mathematical Analysis and Applications*2006,**314**(2):701–709. 10.1016/j.jmaa.2005.04.018MathSciNetView 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 - Deimling K:
**Zeros of accretive operators.***Manuscripta Mathematica*1974,**13:**365–374. 10.1007/BF01171148MathSciNetView ArticleMATHGoogle Scholar - He S, Zhao J, Li Z:
**Degree of convergence of modified averaged iterations for fixed points problems and operator equations.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(9):4098–4104. 10.1016/j.na.2009.02.102MathSciNetView ArticleMATHGoogle Scholar - Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar - Morales C:
**Pseudocontractive mappings and the Leray-Schauder boundary condition.***Commentationes Mathematicae*1979,**20**(4):745–756.MathSciNetMATHGoogle Scholar - Morales CH, Jung JS:
**Convergence of paths for pseudocontractive mappings in Banach spaces.***Proceedings of the American Mathematical Society*2000,**128**(11):3411–3419. 10.1090/S0002-9939-00-05573-8MathSciNetView ArticleMATHGoogle Scholar - Osilike MO:
**Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps.***Journal of Mathematical Analysis and Applications*2004,**294**(1):73–81. 10.1016/j.jmaa.2004.01.038MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Strong convergence theorems for resolvents of accretive operators in Banach spaces.***Journal of Mathematical Analysis and Applications*1980,**75**(1):287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Applications*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Xu H-K, Ori RG:
**An implicit iteration process for nonexpansive mappings.***Numerical Functional Analysis and Optimization*2001,**22**(5–6):767–773. 10.1081/NFA-100105317MathSciNetView ArticleMATHGoogle Scholar - Yao Y, Liou Y-C, Chen R:
**Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(12):3311–3317. 10.1016/j.na.2006.10.013MathSciNetView 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:
**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

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