Abstract
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.
Keywords:
hierarchical fixed point; nonexpansive mapping; Lipschitzian and strongly monotone mapping; quadratic minimization; modified iterative projection algorithm1 Introduction
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 LLipschitzian 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 ηstrongly 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 Cconditions, a family of finite or infinite nonexpansive mappings and pseudocontractions and obtained many useful results; see, for example, [112] 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 [15] 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 [16] introduced the general steepestdescent method
where F is an LLipschitzian and ηstrongly 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.[17] investigated the following iterative method:
where U is a Lipschitzian (possibly nonself) 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.[18] 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 γLipschitzian (possibly nonself) mapping, F is an LLipschitzian and ηstrongly 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 .
Obviously, (2), (3), (4) and (6) are some special cases of (7), respectively. So, our results improve and extend many recent corresponding results of other authors such as [5,13,1519].
2 Preliminaries
This section contains some lemmas which will be used in the proofs of our main results in the following section.
Lemma 2.1[18]
Letandbe any points. The following results hold.
(1) is nonexpansive andif and only if the following relation holds:
(2) if and only if the following relation holds:
Lemma 2.2[4]
LetHbe a real Hilbert space, , the following inequality holds:
Lemma 2.3[17]
Letbe aγLipschitzian mapping with a constantand letbe akLipschitzian andηstrongly monotone mapping with constants, then for,
That is to say, the operatorisstrongly monotone.
Lemma 2.4[10] (Demiclosedness principle)
Let Ω be a nonempty closed convex subset of a real Hilbert spaceHand letbe a nonexpansive mapping with. Ifis a sequence in Ω weakly converging toxandconverges strongly toy, then. In particular, if, then.
Lemma 2.5[19]
Suppose thatand. Letbe anLLipschitzian andηstrongly monotone operator with constants. In association with a nonexpansive mapping, define the mappingby
Thenis a contraction provided, that is,
Lemma 2.6[20]
Letbe a sequence of nonnegative real numbers satisfying the following relation:
with conditions
3 Main results
Theorem 3.1Let Ω be a nonempty closed convex subset of a real Hilbert spaceHand letbe any given initial guess. Letbe nonexpansive mappings such that. Letbe anLLipschitzian andηstrongly monotone (possibly nonself) operator with coefficients. Letbe aγLipschitzian (possibly nonself) mapping with a coefficient. Suppose the parameters satisfy, , where. And suppose the sequencessatisfy the following conditions:
Then the sequencegenerated by (7) converges strongly to a fixed pointofT, which is the unique solution of the variational inequality (5). In particular, if we take, , thendefined by (7) converges in norm to the minimum norm fixed pointof T, namely, the pointis 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
Hence
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
Hence
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.
Remark 3.2 Our Theorem 3.1 improves and extends many recent corresponding main results of other authors (see, for example, [5,13,1519]) in the following ways:
(a) Some selfmappings in other papers (see [15,16,19]) are extended to the cases of nonselfmappings. Such as the selfcontraction mapping in [15,16,19] is extended to the case of a Lipschitzian (possibly nonself)mapping on a nonempty closed convex subset C of H. The Lipschitzian and strongly monotone (self)mapping in [16] is extended to the case of a Lipschitzian and strongly monotone (possibly nonself)mapping .
(b) The contractive mapping f with a coefficient in other papers (see [15,16,18,19]) is extended to the cases of the Lipschitzian mapping U with a coefficient constant .
(c) The Manntype iterative format in [1517,19] has been extended to the Ishikawatype 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.
(d) 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 [19]. If the Lipschitzian mapping , , , we get the variational inequality , , which is the variational inequality studied by Yao et al.[18]. So, the results of Theorem 3.1 in this paper have many useful applications such as the quadratic minimization problem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.
Acknowledgements
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).
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