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Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and variational inequalities
Fixed Point Theory and Applications volume 2013, Article number: 121 (2013)
This article aims to deal with a new modified iterative projection method for solving a hierarchical fixed point problem. It is shown that under certain approximate assumptions of the operators and parameters, the modified iterative sequence converges strongly to a fixed point of T, also the solution of a variational inequality. As a special case, this projection method solves some quadratic minimization problem. The results here improve and extend some recent corresponding results by other authors.
MSC:47H10, 47J20, 47H09, 47H05.
Let Ω be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm . Recall that a mapping is called L-Lipschitzian if there exits a constant L such that , . In particular, if , then T is said to be a contraction; if , then T is called a nonexpansive mapping. We denote by the set of the fixed points of T, i.e., .
A mapping is called η-strongly monotone if there exists a constant such that
In particular, if , then F is said to be monotone.
A mapping is called a metric projection if there exists a unique nearest point in Ω denoted by such that
Recently many authors investigated the fixed point problem of nonexpansive mappings, generalized nonexpansive mappings with C-conditions, a family of finite or infinite nonexpansive mappings and pseudo-contractions and obtained many useful results; see, for example, [1–12] and the references therein.
Now, we focus on the following problem.
To find a hierarchical fixed point of T with respect to another operator S is to find an satisfying
which is equivalent to the following fixed point problem: to find an that satisfies . We know that is closed and convex, so the metric projection is well defined.
It is well known that the iterative methods for finding hierarchical fixed points of nonexpansive mappings can also be used to solve a convex minimization problem; see, for example, [13, 14] and the references therein. In 2006, Marino and Xu  considered the following general iterative method:
where f is a contraction, T is a nonexpansive mapping, A is a bounded linear strongly positive operator: , , for some . And it is proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (2) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
where h is a potential function for γf, i.e., , .
In 2010, Tian  introduced the general steepest-descent method
where F is an L-Lipschitzian and η-strongly monotone operator. Under certain approximate conditions, the sequence generated by (3) converges strongly to a fixed point of T, which solves the variational inequality
Very recently, Ceng et al.  investigated the following iterative method:
where U is a Lipschitzian (possibly non-self) mapping, and F is a Lipschitzian and strongly monotone mapping. They proved that under some approximate assumptions on the operators and parameters, the sequence generated by (4) converges strongly to the unique solution of the variational inequality
On the other hand, in 2010, Yao et al.  investigated an iterative method for a hierarchical fixed point problem by
where is a nonexpansive mapping. Under some approximate assumptions of the parameters, the sequence generated by (6) converges strongly to the unique solution of the variational inequality
Motivated and inspired by the above research work, we introduce the following modified iterative method for a hierarchical fixed point problem:
where S, T are nonexpansive mappings with , U is a γ-Lipschitzian (possibly non-self) mapping, F is an L-Lipschitzian and η-strongly monotone operator. We prove that the sequence generated by (7) converges strongly to the unique solution of the variational inequality (5) if the operators and parameters satisfy some approximate conditions. As a special case, this projection method also solves the quadratic minimization problem .
This section contains some lemmas which will be used in the proofs of our main results in the following section.
Lemma 2.1 
Let and be any points. The following results hold.
is nonexpansive and if and only if the following relation holds:
if and only if the following relation holds:
Lemma 2.2 
Let H be a real Hilbert space, , the following inequality holds:
Lemma 2.3 
Let be a γ-Lipschitzian mapping with a constant and let be a k-Lipschitzian and η-strongly monotone mapping with constants , then for ,
That is to say, the operator is -strongly monotone.
Lemma 2.4  (Demiclosedness principle)
Let Ω be a nonempty closed convex subset of a real Hilbert space H and let be a nonexpansive mapping with . If is a sequence in Ω weakly converging to x and converges strongly to y, then . In particular, if , then .
Lemma 2.5 
Suppose that and . Let be an L-Lipschitzian and η-strongly monotone operator with constants . In association with a nonexpansive mapping , define the mapping by
Then is a contraction provided , that is,
Lemma 2.6 
Let be a sequence of nonnegative real numbers satisfying the following relation:
3 Main results
Theorem 3.1 Let Ω be a nonempty closed convex subset of a real Hilbert space H and let be any given initial guess. Let be nonexpansive mappings such that . Let be an L-Lipschitzian and η-strongly monotone (possibly non-self) operator with coefficients . Let be a γ-Lipschitzian (possibly non-self) mapping with a coefficient . Suppose the parameters satisfy , , where . And suppose the sequences satisfy the following conditions:
Then the sequence generated by (7) converges strongly to a fixed point of T, which is the unique solution of the variational inequality (5). In particular, if we take , , then defined by (7) converges in norm to the minimum norm fixed point of T, namely, the point is the unique solution to the quadratic minimization problem .
Proof We divide the proof into six steps.
Step 1. We first show that the variational inequality (5) has only one solution. Observe that the constants satisfy and
therefore the operator is -strongly monotone, and we get the uniqueness of the solution of the variational inequality (5) and denote it by .
Step 2. Then we get that the sequences and are bounded. By condition (ii), without loss of generality, we may assume , . Taking a fixed point , we have
On the other hand, denoting , from (7) we get
Together with (8) and (9), we have
We get the sequence is bounded, and so are , , , .
Step 3. Next we show that as . Estimate
where M is a constant such that
Substituting (10) into (11), we obtain
Notice the conditions (i) and (iii), by Lemma 2.6, we have as .
Step 4. Next we show that as .
Notice that , , and are bounded, and we have as .
Step 5. Now we show that , where is the unique solution of the variational inequality. Since is bounded, we take a subsequence of such that
and we assume . By Lemma 2.4, we have . Therefore
On the other hand, taking in (8), we obtain . Together with (12), we have
which implies that
By the conditions (i) and (ii), we have and
According to Lemma 2.6, we have .
Step 6. In particular, if we take , , then , which implies that is the minimum norm fixed point of T and satisfies the variational inequality (5)
So, , we deduce , i.e., the point is the unique solution to the quadratic minimization problem . This completes the proof. □
Remark 3.1 Prototypes for the iteration parameters in Theorem 3.1 are, for example, , (with ). It is not difficult to prove that the conditions (i)-(iii) are satisfied.
Some self-mappings in other papers (see [15, 16, 19]) are extended to the cases of non-self-mappings. Such as the self-contraction mapping in [15, 16, 19] is extended to the case of a Lipschitzian (possibly non-self-)mapping on a nonempty closed convex subset C of H. The Lipschitzian and strongly monotone (self-)mapping in  is extended to the case of a Lipschitzian and strongly monotone (possibly non-self-)mapping .
The Mann-type iterative format in [15–17, 19] has been extended to the Ishikawa-type iterative format (7) in our Theorem 3.1. So, their iterative formats (2), (3), (4) and (6) are some special cases of our iterative format (7), and some of their main results have been included in our Theorem 3.1, respectively.
The iterative approximating fixed point of T in Theorem 3.1 is also the unique solution of the variational inequality (5). In fact, (5) is a hierarchical fixed point problem which closely relates to a convex minimization problem. In hierarchical fixed point problem (1), if , then we can get the variational inequality (5). In (5), if , then we get the variational inequality , , which just is the variational inequality studied by Suzuki . If the Lipschitzian mapping , , , we get the variational inequality , , which is the variational inequality studied by Yao et al. . So, the results of Theorem 3.1 in this paper have many useful applications such as the quadratic minimization problem .
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The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the article. This study was supported by the National Natural Science Foundations of China (Grant Nos. 11271330, 11071169 ), the Natural Science Foundations of Zhejiang Province of China (Grant No. Y6100696).
The authors declare that they have no competing interests.
All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.
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Wang, Y., Xu, W. Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and variational inequalities. Fixed Point Theory Appl 2013, 121 (2013). https://doi.org/10.1186/1687-1812-2013-121
- hierarchical fixed point
- nonexpansive mapping
- Lipschitzian and strongly monotone mapping
- quadratic minimization
- modified iterative projection algorithm