## Numerical and Analytical Methods for Variational Inequalities and Related Problems with Applications

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# Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

**Academic Editor:**Hong-Kun Xu

#### Abstract

Let be strictly pseudononspreading mappings defined on closed convex subset of a real Hilbert space . Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality .

#### 1. Introduction

Throughout this paper, we always assume that is a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonlinear mapping. Recall the following definitions.

*Definition 1.1. * is said to be(i)monotone if
(ii)strongly monotone if there exists a constant such that
for such a case, is said to be -strongly-monotone, (iii)inverse-strongly monotone if there exists a constant such that
for such a case, is said to be -inverse-strongly monotone, (iv)-Lipschitz continuous if there exists a constant such that

*Remark 1.2. *Let *, *where is a *-*Lipschitz and *-*strongly monotone operator on with and is a Lipschitz mapping on with coefficient *, **. *It is a simple matter to see that the operator is -strongly monotone over *,* that is:

Following the terminology of Browder-Petryshyn [1], we say that a mapping is(1)-strict pseudocontraction if there exists such that
(2)-strictly pseudononspreading if there exists such that
for all ,(3)nonspreading in [2] if
It is shown in [3] that (1.8) is equivalent to

Clearly every nonspreading mapping is -strictly pseudononspreading. Iterative methods for strictly pseudononspreading mapping have been extensively investigated; see [2, 4–6].

Let be a closed convex subset of , and let be -strictly pseudocontractive mappings on such that . Let and be a sequence in . In [7], Acedo and Xu introduced an explicit iteration scheme called the followting cyclic algorithm for iterative approximation of common fixed points of in Hilbert spaces. They define the sequence cyclically by In a more compact form, they rewrite as where , with (mod ), . Using the cyclic algorithm (1.12), Acedo and Xu [7] show that this cyclic algorithm (1.12) is weakly convergent if the sequence of parameters is appropriately chosen.

Motivated and inspired by Acedo and Xu [7], we consider the following cyclic algorithm for finding a common element of the set of solutions of -strictly pseudononspreading mappings . The sequence generated from an arbitrary as follows: Indeed, the algorithm above can be rewritten as where , ; namely, is one of circularly.

#### 2. Preliminaries

Throughout this paper, we write to indicate that the sequence converges weakly to . implies that converges strongly to . The following definitions and lemmas are useful for main results.

*Definition 2.1. *A mapping is said to be demiclosed, if for any sequence which weakly converges to , and if the sequence strongly converges to , then *. *

*Definition 2.2. * is called demicontractive on , if there exists a constant such that

*Remark 2.3. *Every -strictly pseudononspreading mapping with a nonempty fixed point set is demicontractive (see [8, 9]).

*Remark 2.4 (see [10]). * Let be a -demicontractive mapping on with and for : (A1)-demicontractive is equivalent to
(A2) if .

*Remark 2.5. *According to with being a *-*strictly pseudononspreading mapping, we obtain

Proposition 2.6 (see [2]). *Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping. If , then it is closed and convex. *

Proposition 2.7 (see [2]). * Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping. Then is demiclosed at . *

Lemma 2.8 (see [11]). *Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of which satisfies for all . Also consider the sequence of integers defined by
**
Then is a nondecreasing sequence verifying ; it holds that and we have
*

Lemma 2.9. *Let be a real Hilbert space. The following expressions hold: *(i)*,
*(ii)*. *

Lemma 2.10 (see [6]). *Let be a closed convex subset of a Hilbert space , and let be a -strictly pseudononspreading mapping with a nonempty fixed point set. Let be fixed and define by
**
Then . *

Lemma 2.11. *Assume is a closed convex subset of a Hilbert space . *(a)*Given an integer , assume, is a -strictly pseudononspreading mapping for some , . Let be a positive sequence such that . Suppose that has a common fixed point and . Then,
*(b)*Assuming is a -strictly pseudononspreading mapping for some , , let , . If , then
*

*Proof. * To prove (a), we can assume . It suffices to prove that , where with . Let and write and .

From Lemma 2.9 and taking to deduce that
it follows that
Since , we obtain

This together with (2.10) implies that
Since and , we get which implies which in turn implies that since . Thus, .

By induction, we also claim that with is a positive sequence such that , .

To prove (b), we can assume . Set and , , . Obviously
Now we prove
for all and . If , then ; the conclusion holds. From Lemma 2.10, we can know that . Taking , then
Since , we obtain
Namely, , that is:

By induction, we also claim that the Lemma 2.11(b) holds.

Lemma 2.12. *Let be a closed convex subset of a real Hilbert space , given and . Then if and only if there holds the inequality
*

#### 3. Cyclic Algorithm

In this section, we are concerned with the problem of finding a point such that where , and are -strictly pseudononspreading mappings with , , defined on a closed convex subset in Hilbert space . Here is the set of fixed points of , .

Let be a real Hilbert space, and let be -strongly monotone and -Lipschitzian on with , . Let , . Let be a positive integer, and let be a -strictly pseudononspreading mapping for some , , such that . We consider the problem of finding such that

Since is a nonempty closed convex subset of , VI (3.2) has a unique solution. The variational inequality has been extensively studied in literature; see, for example, [12–16].

*Remark 3.1. *Let be a real Hilbert space. Let be a -Lipschitzian and -strongly monotone operator on with . Leting and leting and , then for and , is a contraction.

*Proof. *Consider
It follows that
So is a contraction.

Next, we consider the cyclic algorithm (1.15), respectively, for solving the variational inequality over the set of the common fixed points of finite strictly pseudononspreading mappings.

Lemma 3.2. *Assume that is defined by (1.15); if is solution of (3.2) with being strictly pseudononspreading mapping and demiclosed and is a bounded sequence such that , then
*

*Proof. *By and demi-closed, we know that any weak cluster point of belongs to . Furthermore, we can also obtain that there exists and a subsequence of such that as (hence ) and
From (3.2), we can derive that
It is the desired result. In addition, the variational inequality (3.7) can be written as
So, by Lemma 2.12, it is equivalent to the fixed point equation

Theorem 3.3. *Let be a nonempty closed convex subset of and for . Let be -strictly pseudononspreading mappings for some , , , and . Let be -Lipschitz mapping on with coefficient , and let be -strongly monotone and -Lipschitzian on with , . Let being a sequence in satisfying the following conditions: *(c1)*,
*(c2)*. **Given , let be the sequence generated by the cyclic algorithm (1.15). Then converges strongly to the unique element in verifying
**
which equivalently solves the variational inequality problem (3.2).*

*Proof. * Take a . Let and . Then , we have

From and (3.11), we also have
Using (1.15) and (3.12), we obtain
which combined with (3.12) and amounts to
Putting , we clearly obtain . Hence is bounded. We can also prove that the sequences and are all bounded.

From (1.15) we obtain that
hence
Moreover, by and using Remark 2.5, we obtain
which combined with (3.16) entails
or equivalently
Furthermore, using the following classical equality
and setting , we have
So (3.19) can be equivalently rewritten as
Now using (3.15) again, we have
Since is -strongly monotone and -Lipschitzian on , hence it is a classical matter to see that
which by and yields
Then from (3.22) and (3.25), we have
The rest of the proof will be divided into two parts.*Case 1*. Suppose that there exists such that is nonincreasing. In this situation, is then convergent because it is also nonnegative (hence it is bounded from below), so that ; hence, in light of (3.26) together with and the boundedness of , we obtain
It also follows from (3.26) that
Then, by , we obviously deduce that
or equivalently (as )
Moreover, by Remark 1.2, we have
which by (3.30) entails
hence, recalling that exists, we equivalently obtain
namely
From (3.27) and Lemma 3.2, we obtain
which yields , so that converges strongly to .*Case 2*. Suppose there exists a subsequence of such that for all . In this situation, we consider the sequence of indices as defined in Lemma 2.8. It follows that , which by (3.26) amounts to
By the boundedness of and , we immediately obtain
Using (1.15), we have
which together with (3.37) and yields

Now by (3.36) we clearly have
which in the light of (3.31) yields
hence (as and (3.37)) it follows that
From (3.37) and Lemma 3.2, we obtain
which by (3.42) yields , so that . Combining (3.39), we have . Then, recalling that (by Lemma 2.8), we get , so that strongly.

Taking , we know that -strictly pseudononspreading mapping is nonspreading mapping and (mod ), . According to the proof Theorem 3.3, we obtain the following corollary.

Corollary 3.4. *Let be a nonempty closed convex subset of . Let be nonspreading mappings and , . Let be -Lipschitz mapping on with coefficient and let be -strongly monotone and -Lipschitzian on with , . Let be a sequence in satisfying the following conditions: *(c1)*,
*(c2)*. **Given , let be the sequence generated by the cyclic algorithm (1.15). Then converges strongly to the unique element in verifying
**
which equivalently solves the variational inequality problem (3.2).*

#### Acknowledgment

This work is supported in part by China Postdoctoral Science Foundation (Grant no. 20100470783).

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#### Copyright

Copyright © 2012 Bin-Chao Deng 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.