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

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Volume 2012 |Article ID 435676 | https://doi.org/10.1155/2012/435676

Bin-Chao Deng, Tong Chen, Zhi-Fang Li, "Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space", Journal of Applied Mathematics, vol. 2012, Article ID 435676, 15 pages, 2012. https://doi.org/10.1155/2012/435676

# Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

Academic Editor: Hong-Kun Xu
Revised01 Jun 2012
Accepted01 Jun 2012
Published18 Jul 2012

#### 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 overโโ,โโ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, 4โ6].

Let be a closed convex subset of , and let be โโ-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, [12โ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.
Caseโโ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 .
Caseโโ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 © 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.

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