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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 218341, 27 pages
Parallel and Cyclic Algorithms for Quasi-Nonexpansives in Hilbert Space
1School of Management, Tianjin University, Tianjin 300072, China
2School of Economics and Management, Shandong University of Science and Technology, Qingdao 266510, China
Received 29 May 2012; Revised 28 July 2012; Accepted 11 August 2012
Academic Editor: Yeong-Cheng Liou
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.
Let be N quasi-nonexpansive mappings defined on a closed convex subset C of a real Hilbert space H. Consider the problem of finding a common fixed point of these mappings and introduce the parallel and cyclic algorithms for solving this problem. We will prove the strong convergence of these algorithms.
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.(1)is said to be monotone if (2) is said to be strongly positive if there exists a constant such that (3) is said to be strongly monotone if there exists a constant such that For such a case, is said to be -strongly monotone.(4) is said to be inverse strongly if there exists a constant such that For such a case, is said to be -inverse-strongly-monotone (-ism).
Assume is strongly positive operator, that is, there is a constant with the property
Remark 1.1. Let , where is strongly positive operator, and is contraction mapping with coefficient . It is a simple matter to see that the operator is -strongly monotone over , that is,
The classical variational inequality which is denoted by is to find such that
The variational inequality has been extensively studied in literature; see, for example, [1, 2] and the reference therein. A mapping is said to be a strict pseudocontraction  if there exists a constant such that for all (If (1.8) holds, we also say that is a -strict pseudo-contraction). These mappings are extensions of nonexpansive mappings which satisfy the inequality (1.8) with . That is, is nonexpansive if
In , Xu proved that the sequence defined by the iterative method below with the initial guess chosen arbitrarily, where the sequence satisfies certain conditions, he proved the sequence converges strongly to the unique solution of the following minimization problem: In , Marino and Xu considered the following general iterative method: they proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.12) converges strongly to the unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ). Some people also study the applications of the iterative method (1.12) [6, 7].
Acedo and Xu  consider the following parallel and cyclic algorithms:
The sequence was generated by where are strict pseudocontractions defined on a closed convex subset of a Hilbert space . Under the following assumptions on the sequences of the weights : (a1) for all and , for all , (a2).
By (1.15), they will prove the weak convergence to a solution of the problem .
They define the sequence cyclically by In a more compact form, they are rewritten as where are -strict pseudo-contractions and with (mod ), . They show that this cyclic algorithm (1.17) is weakly convergent if the sequence of parameters is appropriately chosen. On the other hand, Osilike and Shehu  also consider the cyclic algorithm (1.17), under appropriate assumptions on the sequences of , some strong convergence theorems are proved.
In this paper, we are concerned with the problem of finding a point such that where , and are quasi-nonexpansive mappings defined on a closed convex subset of a Hilbert space . Here is the set of fixed points of , .
Let be defined by where for all such that . Motivated and inspired by Acedo and Xu , we consider the following two general iterative algorithms for a family of quasi-nonexpansive mappings.
Algorithm 1.3. In (1.20), the weights are constant in the sense that they are independent of , the number of steps of the iteration process. In (1.21), we consider a more general case by allowing the weights . Under appropriate assumptions on the sequences of the wights and . From (1.20) and (1.21), we will prove some strong convergence to a solution of the problem (1.18). In addition, we can also know that the condition in  is superfluous.
Another approach to the problem (1.18) is the cyclic algorithm (for convenience, we relabel the mappings as ). This means that beginning with an , we define the sequence cyclically by In a more compact from, can be written as where , , , with (mod ), .
We will show that this cyclic algorithm (1.24) is also strongly convergent if the sequence of parameters is appropriately chosen.
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.
Remark 2.2. From the above definitions, It is easy to see that(i)a nonexpansive mapping is a quasi-nonexpansive mapping; (ii)the set of fixed points of is the set . We assume that , it is well know that is closed and convex.
Remark 2.3 (see ). Let , where is a quasi-nonexpansive on , and . Then the following statements are reached: (i); (ii) is quasi-nonexpansive; (iii), for all ; (iv), for all .
Example 2.4. Let with the norm defined by
and . Then is a nonempty subset of .
Now, for any , define a mapping as follows: It is easy to see that is a quasi-nonexpansive mapping. In fact, for any , taking , that is, we have and
Lemma 2.5. Assume is a closed convex subset of a Hilbert space . (i)Given an integer , assume, for all , is a quasi-nonexpansive. Let be a positive sequence such that . Then is a quasi-nonexpansive.(ii)Let and be given as in (i) above. Suppose that has a common fixed point. Then (iii)Assume be quasi-nonexpansives, let , . If , then
Proof. To prove (i) we only need to consider the case of (the general case can be proved by induction). Set , where and for , is a quasi-nonexpansive. We verify directly the following inequality: for all ,
that is, is a quasi-nonexpansive.
To prove (ii) again we can assume . It suffices to prove that , where with . Let .
Taking to deduce that By the strict convexity of , it follows that ; that is, , hence . According to induction, we can easily claim that (2.6) is holds.
To prove (iii) by induction, for , set for all . Obviously Now we prove For all , , if , then , the conclusion holds. In fact, we can claim that . From Remark 2.3, we know that is quasi-nonexpansive and . Take , then From (2.12), we have namely, , that is,
Suppose that the conclusion holds for , we prove that It suffices to verify for all , that is, . Using Remark 2.3 again, take , we obtain From (2.17), we obtain this implies that namely, From (2.20) and inductive assumption, we have therefore Substituting it into (2.20), we obtain . Thus we assert that
Definition 2.6. A mapping is said to be demiclosed, if for any sequence weakly converges to , and if the sequence strongly converges to , then .
Lemma 2.7 (see ). Assume is a strong positive linear bounded operator on a Hilbert space with coefficient and , then .
Lemma 2.8 (see ). Let be a Hilbert space, a closed convex subset of , and a nonexpansive mapping with , if is a sequence in weakly converging to and if converges strongly to , then .
Lemma 2.9 (see ). 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 , for all , it holds that and one has
Lemma 2.10. Let be a closed convex subset of a real Hilbert space , given and . Then if and only if there holds the inequality
3. Parallel Algorithm
In this section, we discuss the parallel algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite quasi-nonexpansives.
Before stating our main convergence result, we establish the boundedness of the iterates given by following algorithm:
In (3.1), the weight are constant in the sense that they are independent of , the number of steps of the iteration process. Below we consider a more general case by allowing the weights to be step dependent. That is, initializing with , we define by the algorithm From (3.1) and (3.2), the sequence which converges strongly to the unique solution of variational inequality problem : find in such that or equivalently where denotes the metric projection from onto (see,  for more details on the metric projection).
Lemma 3.1. The sequence is generated by (3.2), where and are sequence in , and is a quasi-nonexpansive mapping on , is bounded and satisfies where is any element in , .
Proof. Since , we shall assume that and . Observe that if , then By Lemma 2.7, we obtain Let , for all . By Lemma 2.5, each is a quasi-nonexpansive mapping on , and in light of Remark 2.3. Taking , we have From (3.1), we have By simple inductions, we obtain which gives that the sequence is bounded.
Proof. Clearly, by and demi-closed, we know that any weak cluster point of belongs to . It is also a simple matter to see that there exist and a subsequence of such that (hence ) and such that it follows from (3.3), we can derive that that is the desired result.
Theorem 3.3. Let be a closed convex subset of a Hilbert space and let be a quasi-nonexpansive for , , such that , be a contraction with coefficient , and a positive constant such that for all and for all . Let be a strongly positive bounded linear operator with coefficient . Given the initial guess chosen arbitrarily and given sequences and in , satisfying the following conditions: (c1) and , (c2).
Let be the sequence generated by (3.2). Then converges strongly to the unique a element verifying which equivalently solves the following variational inequality problem:
Proof. Taking , for all . By Lemma 2.5(i), each is a quasi-nonexpansive mapping on , and (3.2) can be rewritten as
Denote by the common fixed point of the mappings (by Lemma 2.5(ii), we can easily know that and take and from (3.16) we deduce that
Moreover, by and using Remark 2.3(iv), we obtain
which combined with the (3.18) entails
Furthermore, using the following classical equality:
and setting , we have
So that (3.21) can be equivalently rewritten as
Now using (3.16) again, we have
Since is a strongly positive bounded linear operator with coefficient , hence it is a classical matter to see that
Then from (3.24) and (3.28), 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 (3.29) together with , , and the boundedness of , we obtain By (3.27) and (3.30), we can easily claim that It also follows from (3.29) that Then, by , we obviously deduce that Since and are both bounded, and , we obtain Moreover, by Remark 1.1, we have which by (3.34) entails hence, recalling that exists, we equivalently obtain namely, From (3.30) and invoking Lemma 3.2, we obtain which by (3.38) 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 2.9. It follows that , which by (3.29) amounts to hence, by the boundedness of and , we immediately obtain From (3.28) we have which together with (3.41), and yields Now by (3.40), we clearly have which in the light of (3.38) yields hence (as ) it follows that From (3.41) and invoking Lemma 3.2, we obtain which by (3.46) yields , so that . Combining (3.43), we have . Then, recalling that (by Lemma 2.9), we get , so that strongly. In addition, the variational inequality (3.39) and (3.47) can be written as So, by the Lemma 2.10, it is equivalent to the fixed point equation
Corollary 3.4. Let be a closed convex subset of a Hilbert space and let be a quasi-nonexpansive for , , such that , is a contraction with coefficient and is a positive constant such that . Let be a strongly positive bounded linear operator with coefficient . Given the initial guess chosen arbitrarily and given sequences and in , satisfying the following conditions: (c1) and ; (c2).
Let be the sequence generated by (3.1). Then converges strongly to the unique a element verifying which equivalently solves the following variational inequality problem:
4. Cyclic Algorithm
In this section, we discuss the cyclic algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite quasi-nonexpansives and introduce quasi-shrinking mapping and quoted its definition from . Hereafter, for nonempty closed set and , we use the notations , , and . In this case, by the upper semicontinuity of (see e.g., [14, Theorem 1.3.3]), is closed. Moreover, for a nonempty bounded closed convex set and , it is not hard to verify that (i) and are also closed; (ii) and are bounded; (iii) is convex.
Definition 4.1 (see ). Suppose that is quasi-nonexpansive with for some closed convex set . Then is called quasi-shrinking on if satisfies . In particular, if is quasi-shrinking on , then is just called quasi-shrinking.
Let be a closed convex subset of a Hilbert space and let be quasi-nonexpansives defined on such that the common fixed point set where , . Let , let , and sequences in . The cyclic algorithm generates a sequence in the following way: In general, is defined by where , with (mod ), .
Lemma 4.2 (see ). Let satisfy (i), (ii).
Let satisfy . Then any sequence satisfying converges to .
Lemma 4.3 (see ). Assume that is a sequence of nonnegative real numbers such that
where and satisfy the following conditions: (i) and , (ii) or .
Theorem 4.4. Let be a closed convex subset of a Hilbert space and let be quasi-nonexpansives for , , such that and a contraction with coefficient . Let be a strongly positive bounded linear operator with coefficient . Given the initial guess chosen arbitrarily and given sequences and in , satisfying the following conditions: (4.1a) and ; (4.1b) or ; (4.1c).
Let be the sequence generated by (4.4). Then converges strongly to the unique a element verifying which equivalently solves the following variational inequality problem:
Proof. Take a . We break the proof process into several steps.
Step 1. is bounded. In light of the Remark 2.3, we obtain From (4.4), we have By simple inductions, we obtain which gives that the sequence is bounded; we also know that and are bounded.
Step 2. Moreover if is quasi-shrinking on the set , we obtain the following statements: (a); (b)