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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 315835, 13 pages

http://dx.doi.org/10.1155/2012/315835

## Computing the Fixed Points of Strictly Pseudocontractive Mappings by the Implicit and Explicit Iterations

Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 16 February 2012; Accepted 1 March 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Yeong-Cheng Liou. 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

It is known that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings have been constructed in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

#### 1. Introduction

In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Our motivations are mainly in two respects.

*Motivation 1*

Iterative methods for finding fixed points of nonexpansive mappings have received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see [1–35] and the references therein. It is known [36] that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. Therefore it is interesting to develop the algorithms for strictly pseudocontractive mappings.

*Motivation 2*

In many problems, it is needed to find a solution with minimum norm. In an abstract way, we may formulate such problems as finding a point with the property
where is a nonempty closed convex subset of a real Hilbert space . A typical example is the least-squares solution to the constrained linear inverse problem [37]. Some related works for finding the minimum-norm solution (or fixed point of nonexpansive mappings) have been considered by some authors. The reader can refer to [38–41].

In the present paper, we present two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

#### 2. Preliminaries

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of .

##### 2.1. Some Concepts

Recall that a mapping is called nonexpansive, if for all . And a mapping is said to be strictly pseudocontractive if there exists a constant such that for all . For such a case, we also say that is a -strictly pseudocontractive mapping. It is clear that, in a real Hilbert space , (2.2) is equivalent to for all . It is clear that the class of strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.

Recall that the nearest point (or metric) projection from onto is defined as follows: for each point , is the unique point in with the property: Note that is characterized by the inequality: Consequently, is nonexpansive.

##### 2.2. Several Useful Lemmas

Lemma 2.1 (see [42]). *Let be a real Hilbert space. There holds the following identity:
**
for all .*

Lemma 2.2 (see [43]). * Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strict pseudocontraction. Then, *(i)* is closed convex so that the projection is well defined; *(ii)* for , is nonexpansive. *

Lemma 2.3 2.3 (see [42]). *Let be a nonempty closed convex of a real Hilbert space . Let be a -strictly pseudocontractive mapping. Then is demiclosed at that is if and , then .*

Lemma 2.4 (see [44]). *Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that
**
for all and
**
Then .*

Lemma 2.5 (see [45]). * Let be a sequence of nonnegative real numbers satisfying
**
where and satisfy *(i)*,*(ii)*either or . ** Then converges to .*

We use the following notation: (i) stands for the set of fixed points of ; (ii) stands for the weak convergence of to ; (iii) stands for the strong convergence of to .

#### 3. Iterations and Convergence Analysis

Theorem 3.1. *Let be a nonempty closed convex subset of a real Hilbert space . Let a -strictly pseudocontractive mapping with . Let be a constant. For and any , let be the sequence defined by the following implicit manner:
**
Then the sequence converges strongly to .*

* Proof. **Step *1. The sequence is well defined.

Set . It is easily to check that . Then, we can rewrite (3.1) as
which is equivalent to the following:
Note that is nonexpansive (see Lemma 2.2). For fix , we define a mapping by
For , we have
which implies that is a self-contraction of for every . Hence has a unique fixed point which is the unique solution of the fixed point equation (3.3).*Step *2. The sequence is bounded.

Pick up any . From (3.3), we have
It follows that

Hence, is bounded and so is .* Step *3. .

From (3.3), we have

It follows that
*Step* 4. .

Since is bounded, there exists a subsequence of , which converges weakly to a point . Noticing (3.9) we can use Lemma 2.3 to get .

By using the convexity of the norm and Lemma 2.1, for any , we have
It turns out that
where is some constant such that
Therefore we can substitute for in (3.11) to get
However, . This together with (3.13) guarantees that . It is clear that . As a matter of fact, in (3.11), if we let , then we get
This is equivalent to
Hence, . Therefore, . This completes the proof.

Corollary 3.2. *Let be a nonempty closed convex subset of a real Hilbert space . Let a nonexpansive mapping with . Let be a constant. For and any , let be the sequence defined by the following implicit manner:
**
Then the sequence converges strongly to .*

Corollary 3.3. *Let be a nonempty closed convex subset of a real Hilbert space . Let a -strictly pseudocontractive mapping with . Let be a constant. For any , let be the sequence defined by the following implicit manner:
**
Then the sequence converges strongly to which is the minimum norm fixed point of .*

Corollary 3.4. *Let be a nonempty closed convex subset of a real Hilbert space . Let a nonexpansive mapping with . Let be a constant. For any , let be the sequence defined by the following implicit manner:
**
Then the sequence converges strongly to which is the minimum norm fixed point of .*

Next, we introduce an explicit algorithm for finding the fixed point of .

Theorem 3.5. *Let be a nonempty closed convex subset of a real Hilbert space . Let a -strictly pseudocontractive mapping with . Let and be two constants in satisfying . For and any , let be the sequence defined by the following explicit manner:
**
where satisfies the following conditions: *)*, *()*. ** Then the sequence converges strongly to .*

* Proof. ** Step *1. The sequence is bounded.

First, we can rewrite (3.19) as
Take . From (3.20), we have
By induction,
Hence, the sequence is bounded and is also bounded.* Step *2*. *.

We can rewrite (3.20) as
where
It follows that
Thus,
This together with Lemma 2.4 implies that
Note that
It follows that
Thus,
*Step *3. , where .

To see this, we can take a subsequence of satisfying the properties
By the demiclosed principle (see Lemma 2.3) and (3.30), we have that . So,
*Step *4. .

From (3.20), we get
where and
It is easy to see that and . We can therefore apply Lemma 2.5 to (3.34) and conclude that as . This completes the proof.

Corollary 3.6. *Let be a nonempty closed convex subset of a real Hilbert space . Let a nonexpansive mapping with . Let and be two constants in satisfying . For and any , let be the sequence defined by the following explicit manner:
**
where satisfies the following conditions: *()*, *()*. ** Then the sequence converges strongly to .*

Corollary 3.7. *Let be a nonempty closed convex subset of a real Hilbert space . Let a -strictly pseudocontractive mapping with . Let and be two constants in satisfying . For any , let be the sequence defined by the following explicit manner:
**
where satisfies the following conditions: *()*, *()*. ** Then the sequence converges strongly to which is the minimum norm fixed point of .*

Corollary 3.8. *Let be a nonempty closed convex subset of a real Hilbert space . Let a nonexpansive mapping with . Let and be two constants in satisfying . For any , let be the sequence defined by the following explicit manner:
**
where satisfies the following conditions: *()*; *()*. ** Then the sequence converges strongly to which is the minimum norm fixed point of .*

#### 4. Conclusion

Finding fixed points of nonlinear mappings (especially, nonexpansive mappings) has received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery and signal processing. It is wellknown that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to construct the methods for computing the fixed points of strictly pseudocontractive mappings. Two iterative methods have been presented. Especially, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings. The ideas contained in the present paper can help us to solve the minimum norm problems in the applied science.

#### Acknowledgment

The author was supported in part by NSC 100-2221-E-230-012.

#### References

- F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,”
*Archive for Rational Mechanics and Analysis*, vol. 24, pp. 82–90, 1967. View at Google Scholar - B. Halpern, “Fixed points of nonexpanding maps,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 957–961, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 591–597, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. L. Lions, “Approximation de points fixes de contractions,” vol. 284, no. 21, pp. A1357–A1359, 1977. View at Google Scholar · View at Zentralblatt MATH
- K. Goebel and W. A. Kirk,
*Topics in Metric Fixed Point Theory*, vol. 28 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar - R. Wittmann, “Approximation of fixed points of nonexpansive mappings,”
*Archiv der Mathematik*, vol. 58, no. 5, pp. 486–491, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Reich and A. J. Zaslavski, “Convergence of Krasnoselskii-Mann iterations of nonexpansive operators,”
*Mathematical and Computer Modelling*, vol. 32, no. 11–13, pp. 1423–1431, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. T. M. Lau and W. Takahashi, “Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces,”
*Nonlinear Analysis*, vol. 70, no. 11, pp. 3837–3841, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. L. Combettes and T. Pennanen, “Generalized Mann iterates for constructing fixed points in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 275, no. 2, pp. 521–536, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 202, no. 1, pp. 150–159, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Moudafi, “Viscosity approximation methods for fixed-points problems,”
*Journal of Mathematical Analysis and Applications*, vol. 241, no. 1, pp. 46–55, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - S. A. Hirstoaga, “Iterative selection methods for common fixed point problems,”
*Journal of Mathematical Analysis and Applications*, vol. 324, no. 2, pp. 1020–1035, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Petruşel and J.-C. Yao, “Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings,”
*Nonlinear Analysis*, vol. 69, no. 4, pp. 1100–1111, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triplehierarchical constrained optimization problem,”
*Mathematical & Computer Modelling*, vol. 55, no. 3-4, pp. 1506–1515, 2012. View at Publisher · View at Google Scholar - Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,”
*Optimization Letters*, vol. 6, no. 4, pp. 621–628, 2012. View at Publisher · View at Google Scholar - Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article 79, 2011. View at Publisher · View at Google Scholar - S. Reich and H. K. Xu, “An iterative approach to a constrained least squares problem,”
*Abstract and Applied Analysis*, no. 8, pp. 503–512, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y. C. Liou, and G. Marino, “Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 279058, 7 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. Shahzad, “Approximating fixed points of non-self nonexpansive mappings in Banach spaces,”
*Nonlinear Analysis*, vol. 61, no. 6, pp. 1031–1039, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Zegeye and N. Shahzad, “Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings,”
*Applied Mathematics and Computation*, vol. 191, no. 1, pp. 155–163, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,”
*Taiwanese Journal of Mathematics*, vol. 5, no. 2, pp. 387–404, 2001. View at Google Scholar · View at Zentralblatt MATH - 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 - W. Takahashi and K. Shimoji, “Convergence theorems for nonexpansive mappings and feasibility problems,”
*Mathematical and Computer Modelling*, vol. 32, no. 11–13, pp. 1463–1471, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - 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 - Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,”
*Nonlinear Analysis*, vol. 68, no. 6, pp. 1687–1693, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. C. Ceng, P. Cubiotti, and J. C. Yao, “Strong convergence theorems for finitely many nonexpansive mappings and applications,”
*Nonlinear Analysis*, vol. 67, no. 5, pp. 1464–1473, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Chang, “Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 2, pp. 1402–1416, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y. C. Liou, and G. Marino, “Strong convergence of some algorithms for
*λ*-strict pseudo-contractions in Hilbert spaces,”*Bulletin of Australian Mathematical Society*, vol. 85, no. 2, pp. 232–240, 2012. View at Publisher · View at Google Scholar - Y. Yao and J.-C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,”
*Applied Mathematics and Computation*, vol. 186, no. 2, pp. 1551–1558, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 208692, 13 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Some developments in general variational inequalities,”
*Applied Mathematics and Computation*, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Extended general variational inequalities,”
*Applied Mathematics Letters*, vol. 22, no. 2, pp. 182–185, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Noor, “Some aspects of extended general variational inequalities,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 303569, 16 pages, 2012. View at Publisher · View at Google Scholar - O. Scherzer, “Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems,”
*Journal of Mathematical Analysis and Applications*, vol. 194, no. 3, pp. 911–933, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative least-squares and regularization,”
*IEEE Transactions on Signal Processing*, vol. 46, no. 9, pp. 2345–2352, 1998. View at Publisher · View at Google Scholar - Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,”
*Nonlinear Analysis*, vol. 72, no. 7-8, pp. 3447–3456, 2010. View at Publisher · View at Google Scholar - Y. L. Cui and X. Liu, “Notes on Browder's and Halpern's methods for nonexpansive mappings,”
*Fixed Point Theory*, vol. 10, no. 1, pp. 89–98, 2009. View at Google Scholar - X. Liu and Y. Cui, “The common minimal-norm fixed point of a finite family of nonexpansive mappings,”
*Nonlinear Analysis*, vol. 73, no. 1, pp. 76–83, 2010. View at Publisher · View at Google Scholar - Y. Yao and H. K. Xu, “Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications,”
*Optimization*, vol. 60, no. 6, pp. 645–658, 2011. View at Publisher · View at Google Scholar - 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 - H. Zhou, “Convergence theorems of fixed points for
*λ*-strict pseudo-contractions in Hilbert spaces,”*Nonlinear Analysis*, vol. 69, no. 2, pp. 456–462, 2008. View at Publisher · View at Google Scholar - T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,”
*Fixed Point Theory and Applications*, no. 1, pp. 103–123, 2005. View at Google Scholar · View at Zentralblatt MATH - 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