Abstract and Applied Analysis

Volume 2013 (2013), Article ID 478647, 6 pages

http://dx.doi.org/10.1155/2013/478647

## Modified Mann-Halpern Algorithms for Pseudocontractive Mappings

School of Science, Shenyang University of Technology, Shenyang 110178, China

Received 18 February 2013; Accepted 20 March 2013

Academic Editor: Yisheng Song

Copyright © 2013 Hong-bo Du. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Modified Mann-Halpern algorithms for finding the fixed points of pseudocontractive mappings are presented. Strong convergence theorems are obtained.

#### 1. Introduction

Finding the fixed points of nonlinear operators is an important topic in fixed point theory, due to the fact that many nonlinear problems can be reformulated as fixed point equations of nonlinear mappings. The research of this area dates back to Picard’s and Banach’s time. Now it is well known that the Picard iterates converge to the unique fixed point of whenever is a contraction of a complete metric space. However, if is not a contraction (e.g., nonexpansive), then the Picard algorithm does not converge. Consequently, Mann’s algorithm was constructed by Mann [1] in 1953: There are a large number of papers on Mann’s algorithm in the literature. See [2–5]. Now we know that if is nonexpansive, then Mann’s algorithm converges weakly to a fixed point of . This algorithm however does not converge in the strong topology.

In order to get the strong convergence, the following Halpern’s algorithm was introduced: The interest and importance of Halpern iterative method lie in the fact that strong convergence of the sequence is achieved under certain mild conditions on parameter in a general Banach space. Please refer to [6–12].

In the present paper, we are devoted to find the fixed points of pseudocontractive mappings. For some related works, please see [13–23]. The interest of pseudocontractions lies in their connection with monotone operators. Browder and Petryshyn [24] studied weak convergence of Mann’s algorithm for the class of strict pseudocontractions. But Mann’s algorithm fails to converge for Lipschitzian pseudocontractions [25].

Inspired by the results in the literature, the main purpose of this paper is to construct an iterative method for finding the fixed points of pseudocontractive mappings. Under some mild conditions, strong convergence results are given.

#### 2. Preliminaries

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of . A mapping is called pseudocontractive if A mapping is called -Lipschitzian if there exists such that for all . In this case, if , then is a -contraction.

It is well known that in a real Hilbert space the following inequality holds: for all .

In the present paper, we will use the following notations: (i)we use to denote the set of fixed points of ;(ii) denotes the weak convergence of to ;(iii) denotes the strong convergence of to .

Lemma 1 (see [26]). *Let be a closed convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping. Then is demiclosed at zero.*

Lemma 2 (see [27]). *Let be a sequence of real numbers. Assume does not decrease at infinity; that is, there exists at least a subsequence of such that for all . For every , define an integer sequence as
**
Then as and for all *

Lemma 3 (see [28]). *Assume that is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence such that*(1)*;
*(2)* or .**
Then .*

#### 3. Main Results

Now we present the statement of our algorithm.

*The Modified Mann-Halpern Algorithm*. Let be a nonempty closed convex subset of a real Hilbert space . Let be a pseudocontractive mapping and a -contractive mapping. Let , , and be three real number sequences in . We have the following steps.(1) Initialization:

(2) Mann step: for a given , define a sequence by for all .(3) Halpern step: for a given and , define for all .

In the following, we assume that(i) the mapping is -Lipschitzian;(ii) the sequences , , and satisfy the following conditions –:;;;;.

Now, we prove our main result as follows.

Theorem 4. *Suppose . Then the sequence defined by (11) converges strongly to a fixed point of .*

*Proof. *Since is a -condition, then is a contractive mapping (where is the metric projection). Hence, there exists a unique such that . In the sequel, we will show that the sequence defined by (11) converges strongly to .

From (11), we getIt is well known that there holds the following inequality in Hilbert spaces:
for all and . Hence, we have
We know that is pseudocontractive if and only if satisfies the condition
for all . Since , we have from (15) that
for all .

By using (13) and (16), we obtain
Note that is -Lipschitzian and
From (17), we have
By condition , without loss of generality, we may assume that for all . Then, we have for all . Substituting (19) to (14) and noting condition , we have
Therefore,
It follows from (12) and (21) that
This implies that the sequence is bounded.

From (5) and (11), we have
Note that (19) is equivalent to
Therefore,
It follows that
Since and are bounded, there exists such that . So
Next, we prove two cases.

Assume there exists an integer such that is decreasing for all .

In this case, we know that exists. From (27), we deduce
By conditions and , we have . Thus, from (28), we get
Since is bounded, there exists a subsequence of satisfying
Thus, we use the demiclosed principle of (Lemma 1) to deduce
So
Returning to (25) and using (5) we obtain
It follows that
In Lemma 3, we take , , and . We can check easily that and . Thus, we deduce that .

Assume there exists an integer such that . In this case, we set . Then, we have . Define an integer sequence for all as follows:
It is clear that is a nondecreasing sequence satisfying
for all . From (28), we get
This implies that . Thus, we obtain
Since , we have from (34) that
It follows that
Combining (38) and (40), we have
and hence
From (34), we obtain
It follows that
This together with (42) implies that
Applying Lemma 2 we get
Therefore, . That is, . The proof is completed.

#### 4. Conclusions

It is now well known that Mann’s algorithm fails to converge for Lipschitzian pseudocontractions. Strong convergence of Ishikawa’s algorithm has not been achieved without compactness assumption. In the present paper, modified Mann-Halpern algorithms for finding the fixed points of pseudocontractive mappings are presented. Strong convergence theorems are obtained.

#### References

- W. R. Mann, “Mean value methods in iteration,”
*Proceedings of the American Mathematical Society*, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,”
*Inverse Problems*, vol. 23, no. 4, pp. 1635–1640, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 67, no. 2, pp. 274–276, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. H. Kim and H. K. Xu, “Robustness of Mann's algorithm for nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 327, no. 2, pp. 1105–1115, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,”
*Nonlinear Analysis. Theory, Methods & Applications A. Theory and Methods*, vol. 68, no. 6, pp. 1687–1693, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Suzuki, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 135, pp. 99–106, 2007. View at Google Scholar - P.-E. Maingé, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 469–479, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 125, no. 12, pp. 3641–3645, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 298, no. 1, pp. 279–291, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Lopez, V. Martin, and H.-K. Xu, “Perturbation techniques for nonexpansive mappings with applications,”
*Nonlinear Analysis. Real World Applications*, vol. 10, no. 4, pp. 2369–2383, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Takahashi, N.-C. Wong, and J.-C. Yao, “Two generalized strong convergence theorems of Halpern's type in Hilbert spaces and applications,”
*Taiwanese Journal of Mathematics*, vol. 16, no. 3, pp. 1151–1172, 2012. View at Google Scholar · View at MathSciNet - H. Zhou, “Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces,”
*Nonlinear Analysis. Theory, Methods & Applications A. Theory and Methods*, vol. 70, no. 11, pp. 4039–4046, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”
*Nonlinear Analysis. Theory, Methods & Applications A. Theory and Methods*, vol. 67, no. 7, pp. 2258–2271, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. R. Sahu, H.-K. Xu, and J.-C. Yao, “Asymptotically strict pseudocontractive mappings in the intermediate sense,”
*Nonlinear Analysis. Theory, Methods & Applications A. Theory and Methods*, vol. 70, no. 10, pp. 3502–3511, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”
*Nonlinear Analysis. Theory, Methods & Applications A. Theory and Methods*, vol. 67, no. 7, pp. 2258–2271, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. C. Ceng, A. Petruşel, and J.-C. Yao, “Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings,”
*Applied Mathematics and Computation*, vol. 209, no. 2, pp. 162–176, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense,”
*Journal of Nonlinear and Convex Analysis*, vol. 11, no. 2, pp. 283–308, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 1, pp. 336–346, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Zegeye, N. Shahzad, and M. A. Alghamdi, “Minimum-norm fixed point of pseudocontractive mappings,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 926017, 15 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. Zegeye, N. Shahzad, and T. Mekonen, “Viscosity approximation methods for pseudocontractive mappings in Banach spaces,”
*Applied Mathematics and Computation*, vol. 185, no. 1, pp. 538–546, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Zhou, “Convergence theorems of fixed points for
*λ*-strict pseudo-contractions in Hilbert spaces,”*Nonlinear Analysis. Theory, Methods & Applications A. Theory and Methods*, vol. 69, no. 2, pp. 456–462, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - L.-C. Zeng, N.-C. Wong, and J.-C. Yao, “Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type,”
*Taiwanese Journal of Mathematics*, vol. 10, no. 4, pp. 837–849, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 20, pp. 197–228, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. E. Chidume and S. A. Mutangadura, “An example of the Mann iteration method for Lipschitz pseudocontractions,”
*Proceedings of the American Mathematical Society*, vol. 129, no. 8, pp. 2359–2363, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - Q.-B. Zhang and C.-Z. Cheng, “Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space,”
*Mathematical and Computer Modelling*, vol. 48, no. 3-4, pp. 480–485, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,”
*Set-Valued Analysis*, vol. 16, no. 7-8, pp. 899–912, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet