Abstract and Applied Analysis

Volume 2014, Article ID 791209, 8 pages

http://dx.doi.org/10.1155/2014/791209

## Viscosity Projection Algorithms for Pseudocontractive Mappings in Hilbert Spaces

^{1}School of Science, Tianjin Polytechnic University, Tianjin 300387, China^{2}College of Management and Economics, Tianjin University, Tianjin 300072, China^{3}Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea^{4}Department of Mathematics, Dong-A University, Busan 614-714, Republic of Korea

Received 6 December 2013; Accepted 28 December 2013; Published 9 February 2014

Academic Editor: Chong Li

Copyright © 2014 Xiujuan Pan et al. 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

An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping.

#### 1. Introduction

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

Recall that a mapping is said to be(i)Lipschitz there exists a constant such that for all ; if , then is said to be contractive; if , then is said to be nonexpansive;(ii)pseudocontractive ; ; , for all .

Interest in pseudocontractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling [1], that if is an accretive operator, then the solutions of the equations correspond to the equilibrium points of some evolution systems. This explains the importance, from this point of view, of the improvement brought by the Ishikawa iteration which was introduced by Ishikawa [2] in 1974. The original result of Ishikawa is stated in the following.

Theorem 1. *Let be a convex compact subset of a Hilbert space and let be an -Lipschitzian pseudocontractive mapping with . For any , define the sequence iteratively by
**
for all , where and satisfy the conditions: and . Then the sequence generated by (1) converges strongly to a fixed point of .*

The iteration (1) is now referred to as the Ishikawa iterative sequence. However, we note that is compact subset. Now, we know that strong convergence has not been achieved without compactness assumption on the involved operation or the underlying spaces. A counter example can be found in Chidume and Mutangadura [3].

In order to obtain a strong convergence result for pseudocontractive mappings without the compactness assumption, in [4], Zhou coupled the Ishikawa algorithm with the hybrid technique and presented the following algorithm for Lipschitz pseudocontractive mappings: Zhou proved that the sequence generated by (2) converges strongly to the fixed point of . Further, in [5], Yao et al. introduced the hybrid Mann algorithm as follows.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence in . Let . For and , define a sequence of as follows: Note that, in iterations (2) and (3), we need to compute the half-spaces (and/or ). Very recently, Zegeye et al. [6] further studied the convergence analysis of the Ishikawa iteration (1). They proved ingeniously the strong convergence of the Ishikawa iteration (1). However, we have to assume that the interior of is nonempty. This appears very restrictive since even in with the usual norm, Lipschitz pseudocontractive maps with finite number of fixed points do not enjoy this condition that . For some related works, please refer to [7–19].

On the other hand, we notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset of a Hilbert space and a bounded linear operator , where is another Hilbert space. The -constrained pseudoinverse of , , is then defined as the minimum-norm solution of the constrained minimization problem which is equivalent to the fixed point problem where is the adjoint of , is a constant, and is such that .

It is, therefore, an interesting problem to invent iterative algorithms that can generate sequences which converge strongly to the minimum-norm solution of a given fixed point problem. The purpose of this paper is to solve such a problem for pseudocontractions. More precisely, we will introduce an explicit projection algorithm with viscosity technique for finding the fixed points of a Lipschitzian pseudocontractive mapping. Strong convergence theorem is demonstrated. Consequently, as an application, we can find the minimum-norm fixed point of the pseudocontractive mapping.

#### 2. Preliminaries

Recall that the metric projection satisfies . The metric projection is a typical firmly nonexpansive mapping. The characteristic inequality of the projection is for all , .

In the sequel we will use the following expressions:(i) denotes the set of fixed points of ;(ii) denotes the weak convergence of to ;(iii) denotes the strong convergence of to .

The following lemmas will be useful for the next section.

Lemma 2 (see [20]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with . Then,
*

*Lemma 3 (see [21]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that a mapping is monotone and weakly continuous along segments (i.e., weakly, as , whenever for ). Then the variational inequality
is equivalent to the dual variational inequality
*

*Lemma 4 (see [22]). Assume that the sequence satisfies and where is a sequence in and is a sequence such that and (or ). Then .*

*3. Main Results*

*3. Main Results*

*In order to prove our main result, we need the following proposition.*

*Proposition 5. Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with . Let be a -contraction.For each , let the net be defined by
Then, as , the net converges strongly to a point which solves the following variational inequality:
*

*Proof. *For , define a mapping by
For any , we have
Hence, is a -contraction on with as its unique fixed point. So, is well defined.

Let . From (9), we have
It follows that
Thus, and are bounded.

Again from (9), we get
Let be a sequence such that as . Put . From (15), we have
From (9), we obtain
It follows that
where is a constant such that
In particular, we have
Noting that is bounded, without loss of generality, we assume that . It is obvious that . From (16) and Lemma 2, we deduce . Substitute for in (20) to get
Since , we deduce from (21) that . The net is, therefore, relatively compact, as , in the norm topology.

In (20), letting , we get
Therefore, solves the variational inequality
By Lemma 3, (23) equals its dual variational inequality
This indicates that . That is, is the unique fixed point in of the contraction . So, the entire net converges in norm to as . This completes the proof.

*Corollary 6. Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with .For each , let the net be defined by
Then, as , the net converges strongly to the minimum-norm fixed point of .*

*Proof. *As a matter of fact, in (9), we choose , and then (9) reduces to (25). From Proposition 5, (24) is reduced to
Equivalently,
This implies that
Therefore, is the minimum-norm fixed point of . This completes the proof.

*Algorithm 7. *Let be a nonempty closed subset of a real Hilbert space . Let be a pseudocontraction and let be a -contraction. Let and be two real number sequences. For , we define a sequence iteratively by

*Theorem 8. Let be a nonempty closed convex subset of a real Hilbert space . Let be an -Lipschitzian and pseudocontraction with and a -contraction. Suppose the following conditions are satisfied: (C1) and ;(C2);(C3).Then the sequence generated by the algorithm (29) converges strongly to .*

*Proof. *First, we prove that the sequence is bounded. We will show this fact by induction. According to conditions (C1) and (C2), there exists a sufficiently large positive integer such that
Fix a and take a constant such that

Next, we show that .

Set
Then we have
Since is pseudocontractive, is monotone. So, we have
From (29), (33), and (34), we obtain
It follows that
By (29), we have
Substitute (37) into (36) to obtain
that is,
By induction, we get
which implies that is bounded and so is . Now, we take a constant such that

Set (i.e., is a resolvent of the monotone operator ). We then have that is a nonexpansive self mapping of and .

By Proposition 5, we know that, whenever and , the sequence defined by
converges strongly to the fixed point of (and of as ). Without loss of generality, we may assume that for all .

It suffices to prove that as (for some ). To this end, we rewrite (42) as
By using the property of the metric projection, we have
Note that
Hence, we get
From (42), we have
It follows that
Set
By condition (C2), and , for large enough. Hence, by (46) and (48), we have
By (29), we have
Next, we estimate . Since , . Then, we have

It follows that
Thus,
where the finite constant is given by
Set
Then (54) can be rewritten as
By the conditions (C1)–(C3), we deuce that
From Lemma 4 and (57), we get as . This completes the proof.

*Algorithm 9. *Let be a nonempty closed subset of a real Hilbert space . Let be a pseudocontraction. Let and be two real number sequences. For , we define a sequence iteratively by

*Corollary 10. Let be a nonempty closed convex subset of a real Hilbert space . Let be an Lipschitzian and pseudocontraction with . Suppose the following conditions are satisfied:(C1) and ;(C2);(C3).Then the sequence generated by the algorithm (59) converges strongly to the minimum-norm fixed point of .*

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgment*

*Acknowledgment*

*The first author was supported by the National Natural Science Foundation of China, Grant no. 11226125.*

*References*

*References*

- K. Deimling, “Zeros of accretive operators,”
*Manuscripta Mathematica*, vol. 13, no. 4, pp. 365–374, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Ishikawa, “Fixed points by a new iteration method,”
*Proceedings of the American Mathematical Society*, vol. 44, pp. 147–150, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. E. Chidume and S. A. Mutangadura, “An example on 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 Zentralblatt MATH · View at Scopus - H. Zhou, “Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 343, no. 1, pp. 546–556, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. Yao, Y.-C. Liou, and G. Marino, “A hybrid algorithm for pseudo-contractive mappings,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 71, no. 10, pp. 4997–5002, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. Zegeye, N. Shahzad, and M. A. Alghamdi, “Convergence of Ishikawa's iteration method for pseudocontractive mappings,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 74, no. 18, pp. 7304–7311, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - 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 Scopus - L.-C. Ceng, A. Petruşel, and J.-C. Yao, “Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense,”
*Taiwanese Journal of Mathematics*, vol. 15, no. 2, pp. 587–606, 2011. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - L.-C. Ceng and J.-C. Yao, “Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense,”
*Acta Applicandae Mathematicae*, vol. 115, no. 2, pp. 167–191, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. C. Yao and L. C. Zeng, “Strong convergence of averaged approximants for asymptotically pseudocontractive mappings in Banach spaces,”
*Journal of Nonlinear and Convex Analysis*, vol. 8, pp. 451–462, 2007. View at Google Scholar - L. C. Ceng, Q. H. Ansari, and C. F. Wen, “Implicit relaxed and hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 854297, 14 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. E. Chidume, M. Abbas, and B. Ali, “Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings,”
*Applied Mathematics and Computation*, vol. 194, no. 1, pp. 1–6, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - C. E. Chidume and H. Zegeye, “Iterative solution of nonlinear equations of accretive and pseudocontractive types,”
*Journal of Mathematical Analysis and Applications*, vol. 282, no. 2, pp. 756–765, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - L. Ćirić, A. Rafiq, N. Cakić, and J. S. Ume, “Implicit Mann fixed point iterations for pseudo-contractive mappings,”
*Applied Mathematics Letters*, vol. 22, no. 4, pp. 581–584, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - C. Moore and B. V. C. Nnoli, “Strong convergence of averaged approximants for Lipschitz pseudocontractive maps,”
*Journal of Mathematical Analysis and Applications*, vol. 260, no. 1, pp. 269–278, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. Udomene, “Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 67, no. 8, pp. 2403–2414, 2007. View at Publisher · View at Google Scholar · View at Scopus - 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 Scopus - 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 Scopus - H. Zhou, “Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 70, no. 11, pp. 4039–4046, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. Geobel and W. A. Kirk,
*Topics in Metric Fixed Point Theory*, vol. 28, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990. - X. Lu, H.-K. Xu, and X. Yin, “Hybrid methods for a class of monotone variational inequalities,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 71, no. 3-4, pp. 1032–1041, 2009. View at Publisher · View at Google Scholar · View at Scopus - H. K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 2, pp. 1–17, 2002. View at Google Scholar

*
*