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On a Two-Step Algorithm for Hierarchical Fixed Point Problems and Variational Inequalities
Journal of Inequalities and Applications volume 2009, Article number: 208692 (2009)
A common method in solving ill-posed problems is to substitute the original problem by a family of well-posed (i.e., with a unique solution) regularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters.
1. Introduction and Preliminar Results
A common method in solving ill-posed problems is to substitute the original problem by a family of well-posed (i.e., with a unique solution) regularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters. We will see that choosing appropriate hypotheses on the parameters, we will obtain convergence to the solution of well-posed problems. Changing these assumptions, we will obtain convergence to one of the solutions of a ill-posed problem. The results are situaded on the lines of research of Byrne , Yang and Zhao , Moudafi , and Yao and Liou .
In this paper, we consider variational inequalities of the form
where are nonexpansive mappings such that the fixed points set of () is nonempty and is a nonempty closed convex subset of a Hilbert space . If we denote with the set of solutions of (1.1), it is evident that .
Variational inequalities of (1.1) cover several topics recently investigated in literature as monotone inclusion ( and the references therein), convex optimization , quadratic minimization over fixed point set (see, e.g., [5, 7–10] and the references therein).
It is well known that the solutions of (1.1) are the fixed points of the nonexpansive mapping .
There are in literature many papers in which iterative methods are defined in order to solve (1.1).
Recently, in  Moudafi defined the following explicit iterative algorithm
where and are two sequences in and he proved a weak-convergence's result. In order to obtain a strong-convergence result, Maingé and Moudafi in  introduced and studied the following iterative algorithm
where and are two sequences in .
Let be a contraction with coefficient In this paper, under different conditions on involved parameters, we study the algorithm
and give some conditions which assure that the method converges to a solution which solves some variational inequality.
We will confront the two methods (1.3) and (1.4) later.
We recall some general results of the Hilbert spaces theory and of the monotone operators theory.
For all , there holds the inequality
If is closed convex subset of a real Hilbert space , the metric projection is the mapping defined as follows: for each , is the only point in with the property
Let be a nonempty closed convex subset of a real Hilbert space and let be the metric projection from onto . Given and , if and only if
Lemma 1.3 (see ).
Let be a contraction with coefficient and be a nonexpansive mapping. Then, for all :
(a)the mapping is strongly monotone with coefficient , that is,
(b)the mapping is monotone, that is,
Finally, we conclude this section with a lemma due to Xu on real sequences which has a fundamental role in the sequel.
Lemma 1.4 (see ).
Assume is a sequence of nonnegative numbers such that
where is a sequence in and is a sequence in such that,
2. Convergence of the Two-Step Iterative Algorithm
Let us consider the scheme
As we will see the convergence of the scheme depends on the choice of the parameters and . We list some possible hypotheses on them:
(H1)there exists such that ;
(H3) as and ;
(H9)there exists such that .
Assume that (H1) holds. Then and are bounded.
Let . Then,
So, by induction, one can see that
Of course is bounded too.
Suppose that (H1), (H3) hold. Also, assume that either (H4) and (H5) hold, or (H6) and (H7) hold. Then
(1) is asymptotically regular, that is,
(2)the weak cluster points set .
then, passing to the norm we have
By definition of one obtain that
so, substituting (2.7) in (2.6) we obtain
By Proposition 2.1, we call so we have
So, if (H4) and (H5) hold, we obtain the asymptotic regularity by Lemma 1.4.
If, instead, (H6) and (H7) hold, from (H1) we can write
so, the asymptotic regularity follows by Lemma 1.4 also.
In order to prove (2), we can observe that
By (H1), and (H3) it follows that , as , so that since is asymptotically regular. By demiclosedness principle we obtain the thesis.
Suppose that the hypotheses of Proposition 2.2 hold. Then
To prove we can observe that
The asymptotical regularity of gives the claim.
Moreover, noting that
since as we obtain . In the end follows easily by and .
Suppose (H2) with and (H3). Moreover Suppose that either (H4) and (H5) hold, or (H6) and (H7) hold. If one denote by the unique element in such that , then
(2) as .
First of all, is a contraction, so there exists a unique such that . Moreover, from Lemma 1.2, is characterized by the fact that
Since (H2) implies (H1), thus is bounded. Let be a subsequence of such that
and . Thanks to either ((H4) and (H5)) or ((H6) and (H7)), by Proposition 2.2 it follows that . Then
Now we observe that, by Lemma 1.1
Thus, by Lemma 1.4, as .
Suppose that (H2) with , (H3), (H8), (H9) hold. Then , as , where is the unique solution of the variational inequality
First of all, we show that (2.20) cannot have more than one solution. Indeed, let and be two solutions. Then, since is solution, for one has
Adding (2.21) and (2.22), we obtain
so . Also now the condition (H2) with implies (H1) so the sequence is bounded. Moreover, since (H8) implies (H6) and (H7), then is asymptotically regular. Similarly, by Proposition 2.2, the weak cluster points set of , , is a subset of . Now we have
and denoting by we have
Dividing by in (2.9), one observe that
By Lemma 1.4, we have
so, also is a null sequence as . Fixing , by (2.26) it results
By Lemma 1.3, we obtain that
Now, we observe that
so, since and , as , then every weak cluster point of is also a strong cluster point.
By Proposition 2.2, is bounded, thus there exists a subsequence converging to . For all by (2.26)
Passing to we obtain
which (2.20). Thus, since the (2.20) cannot have more than one solution, it follows that and this, of course, ensures that , as .
Suppose that (H2) holds with . Suppose that (H3), (H8) and (H9) hold. Moreover let be bounded and be a null sequence. Then every is solution of the variational inequality
that is, .
Since (H8) implies (H6) and (H7), by boundedness of , we can obtain its asymptotical regularity as in proof of Proposition 2.2. Moreover, since , as in proof of Proposition 2.2, . With the same notation in proof of Theorem 2.5 we have that
holds for all . So, if and , by (H2), the boundedness of , and of Corollary 2.3 we have
If we change with , , we have
If we choose and (with ), since and it is not difficult to prove that (H8) is satisfied for and (H9) is satisfied if .
It is clear that our algorithm (1.4) is different from (1.3). At the same time, our algorithm (1.4) includes some algorithms in the literature as special cases. For instance, if we take in (1.4), then we get which is well-known as the viscosity method studied by Moudafi  and Xu .
We do not know the rate of convergence of our method. Nevertheless, the rates of convergence of our method (1.4) that generates the sequence and the Mainge-Moudafi method (1.3), seem not comparable. To see this, we consider three examples. In such examples we take , , , , .
In all three examples all the assumptions (that are the same of the Mainge-Moudafi method) are satisfied and the point at which both the sequences and converge is .
Take and . Then
Now , while, for , it results . For instance, we report here some value
However from the 64th iteration onward, becomes quickly very exiguous with respect to . For instance, while .
Take . Then
that is the sequences and are interchanged with respect to the previous example. So this time for and for .
Take , . Then
so this time .
Reassuming, we cannot affirm that our method is more convenient or better than the Mainge-Moudafi method, but only that seems to us that it is the first time that it is introduced a two-step iterative approach to the VIP (1.1). In some case, our method approximates the solution more rapidly than Mainge-Moudafi method, in some other case it happens the contrary and in some other cases, both methods give the same sequence.
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The authors are extremely grateful to the anonymous referees for their useful comments and suggestions. This work was supported in part by Ministero dell'Universitá e della Ricerca of Italy.
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Cianciaruso, F., Marino, G., Muglia, L. et al. On a Two-Step Algorithm for Hierarchical Fixed Point Problems and Variational Inequalities. J Inequal Appl 2009, 208692 (2009). https://doi.org/10.1155/2009/208692
- Variational Inequality
- Iterative Algorithm
- Real Hilbert Space
- Nonempty Closed Convex Subset
- Null Sequence