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Boundary point algorithms for minimum norm fixed points of nonexpansive mappings
Fixed Point Theory and Applications volume 2014, Article number: 56 (2014)
Let H be a real Hilbert space and C be a closed convex subset of H. Let be a nonexpansive mapping with a nonempty set of fixed points . If , then Halpern’s iteration process cannot be used for finding a minimum norm fixed point of T since may not belong to C. To overcome this weakness, Wang and Xu introduced the iteration process for finding the minimum norm fixed point of T, where the sequence , arbitrarily and is the metric projection from H onto C. However, it is difficult to implement this iteration process in actual computing programs because the specific expression of cannot be obtained, in general. In this paper, three new algorithms (called boundary point algorithms due to using certain boundary points of C at each iterative step) for finding the minimum norm fixed point of T are proposed and strong convergence theorems are proved under some assumptions. Since the algorithms in this paper do not involve , they are easy to implement in actual computing programs.
MSC:47H09, 47H10, 65K10.
1 Introduction and preliminaries
Let H be a real Hilbert space with the inner product and the norm , and let C be a nonempty closed convex subset of H. Recall that a mapping is nonexpansive if for all . We use to denote a set of fixed points of T, i.e., . Throughout this article, is always assumed to be nonempty.
For every nonempty closed convex subset K of H, the metric (or nearest point) projection indicated by from H onto K can be defined, that is, for each , is the only point in K such that . It is well known (e.g., see ) that is nonexpansive and a characteristic inequality holds.
Lemma 1.1 Let K be a closed convex subset of a real Hilbert space H. Given and . Then if and only if there holds the relation
Since is a closed convex subset of H, so the metric projection is valid and thus there exists a unique element, denoted by , in such that , that is, . is called a minimum norm fixed point of T. Because the minimum norm fixed point of a nonexpansive mapping is closely related to convex optimization problems, it is favored by people.
An extensive literature on iteration methods for fixed point problems of nonexpansive mappings has been published (for example, see [1–17]). Many iteration processes are often used to approximate a fixed point of a nonexpansive mapping in a Hilbert space or a Banach space. One of them is now known as Halpern’s iteration process  and is defined as follows: take an initial guess arbitrarily and define recursively by
where is a sequence in the interval and u is some given element in C. For Halpern’s iteration process, a classical result is as follows.
If satisfies the conditions:
then the sequence generated by (1.1) converges strongly to a fixed point of T such that , that is,
Now we consider how to get the minimum norm fixed point of T. In the case where , taking in (1.1), we assert by using Theorem 1.2 that generated by (1.1) converges strongly to under conditions (i)-(iii) above. But, in the case where , the iteration process becomes invalid because may not belong to C. In order to overcome this weakness, Wang and Xu  introduced the iteration process
They proved that if satisfies the same conditions in Theorem 1.2, then the sequence generated by (1.2) converges strongly to .
However, it is difficult to implement the iteration process (1.2) in actual computing programs because the specific expression of cannot be obtained, in general.
The purpose of this paper is to propose three new algorithms for finding the minimum norm fixed point of T. The strong convergence theorems are proved under some assumptions. The main advantage of the algorithms in this paper is that they have nothing to do with the metric projection and thus they are easy to implement in actual computing programs. Because the key of our algorithms is replacing a fixed element u in (1.1) by a certain sequence in the boundary of C, they are called boundary point algorithms.
We will use the following notations:
⇀ for weak convergence and → for strong convergence.
denotes the weak ω-limit set of .
means that B is the definition of A.
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 1.3 ()
Let C be a closed convex subset of a real Hilbert space H, and let be a nonexpansive mapping such that . If a sequence in C is such that and , then .
Lemma 1.4 There holds the identity in a real Hilbert space H:
Assume that is a sequence of nonnegative real numbers satisfying the property
If , and satisfy the conditions:
2 Main results
In this section, C is always assumed to be a nonempty closed convex subset of H such that . We use ∂C to denote the boundary of C. In order to give our main results, we first introduce a function by the definition
It is easy to see that and hold for each due to the assumption .
Since our iteration processes will involve the function , it is necessary to explain how to calculate for any given in actual computing programs. In order to get the value for a given , we often need to deal with an algebraic equation. But dealing with an algebraic equation is easier than calculating the metric projection , in general. To illustrate this viewpoint, let us consider the following simple example.
Example 1 Let H be a real Hilbert space. Define a convex function by
where and u are two given points in H such that . Setting , then it is easy to show that C is a nonempty convex closed subset of H such that (note that and ). For a given , we have . In order to get , let , where is an unknown number. Thus we obtain an algebraic equation
Consequently, we get
By the definition of h, we have
Next we give our first iteration process for finding the minimum norm fixed point of T: take arbitrarily and define recursively by
Remark 1 How to implement the iteration process (2.1)? In actual computing programs, we can use the standard Halpern’s iteration process to get from for each . Indeed, taking and is generated inductively by
then, using Theorem 1.2, as . Thus we can take approximately for a sufficiently large integer in actual computing programs.
Geometric intuition seems to encourage us to guess as under some certain assumptions. As a matter of fact, it is true.
Theorem 2.1 If satisfies , then generated by (2.1) converges strongly to .
Proof Noticing the fact that holds for all , we have from (2.1) that
Thus this together with the condition leads to the conclusion. □
Remark 2 Is the condition reasonable? In other words, can we find an example which satisfies this condition? The answer is yes. The following result implies that this condition is not harsh.
Corollary 1 If , then generated by (2.1) converges strongly to .
Proof Obviously, it suffices to verify that if , then . In fact, setting , we have from (2.1) and (2.2) that
This implies that holds. □
Our second iteration process for finding the minimum norm fixed point of T is defined by
where , () and is taken in C arbitrarily.
Remark 3 Equation (2.3) is an implicit iteration process. A natural question is how to get from . Indeed, suppose that we have got , define the mapping by (), then is -contractive and is just its unique fixed point. So we can use Picard’s iteration process
to calculate approximately since as , where can be taken in C arbitrarily, for example, .
Theorem 2.2 Assume that and , then generated by (2.3) converges strongly to .
Proof We first show that is bounded. Indeed, take a to derive that
It follows that
and is bounded, so are . This together with (2.3) implies that (). Thus it follows from Lemma 1.3 that .
Next we show that
Indeed, take a subsequence of such that
without loss of generality, we may assume that . Noticing , we obtain from and Lemma 1.1 that
Finally, we show that (). As a matter of fact, we have by using Lemma 1.4 that
Using Lemma 1.5, we conclude from (2.5) and conditions and that . □
By a similar argument as above, we easily get the following result.
Corollary 2 If and , then generated by (2.3) converges strongly to , where is the range of T.
Proof It suffices to verify that implies . Indeed,
Setting , we have from (2.4) that
Note that , it follows that . □
Finally, we propose an explicit iteration process for finding the minimum norm fixed point of T which is defined by
where , () and is taken in C arbitrarily.
Theorem 2.3 Assume that and satisfy the following conditions:
Then generated by (2.6) converges strongly to .
Proof We first show that is bounded. Indeed, we have by taking arbitrarily that
This means that is bounded, so are .
We next show that . Using (2.6), it follows from a direct calculation that
Using Lemma 1.5, we conclude from conditions (i)-(iv) that . Noticing the boundedness of and and condition (i), we have from (2.6) that . Consequently, . Using Lemma 1.3, we derive that .
Then we show that
As a matter of fact, this is derived by the same argument as in the proof of Theorem 2.3.
Finally, we show that . Using Lemma 1.4 and (2.6), it is easy to verify that
It is easily seen that , by conditions (i) and (ii), and by (2.7). By Lemma 1.5, we conclude that . □
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This work was supported in part by the Fundamental Research Funds for the Central Universities (3122013k004) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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He, S., Yang, C. Boundary point algorithms for minimum norm fixed points of nonexpansive mappings. Fixed Point Theory Appl 2014, 56 (2014). https://doi.org/10.1186/1687-1812-2014-56
- minimum norm fixed point
- nonexpansive mapping
- metric projection
- boundary point algorithm
- Hilbert space