- Research Article
- Open Access
On a Two-Step Algorithm for Hierarchical Fixed Point Problems and Variational Inequalities
© Filomena Cianciaruso et al. 2009
- Received: 5 May 2009
- Accepted: 12 September 2009
- Published: 28 September 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.
- Variational Inequality
- Iterative Algorithm
- Real Hilbert Space
- Nonempty Closed Convex Subset
- Null Sequence
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.
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
Lemma 1.3 (see ).
Let be a contraction with coefficient and be a nonexpansive mapping. Then, for all :
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 ).
where is a sequence in and is a sequence in such that,
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.
Of course is bounded too.
Suppose that (H1), (H3) hold. Also, assume that either (H4) and (H5) hold, or (H6) and (H7) hold. Then
(2)the weak cluster points set .
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
The asymptotical regularity of gives the claim.
Moreover, noting that
since as we obtain . In the end follows easily by and .
(2) as .
Thus, by Lemma 1.4, as .
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)
which (2.20). Thus, since the (2.20) cannot have more than one solution, it follows that and this, of course, ensures that , as .
that is, .
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 .
However from the 64th iteration onward, becomes quickly very exiguous with respect to . For instance, while .
that is the sequences and are interchanged with respect to the previous example. So this time for and for .
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.
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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