- Open Access
Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one
© Kimura and Satô; licensee Springer. 2013
- Received: 25 July 2012
- Accepted: 18 December 2012
- Published: 9 January 2013
In this paper, we deal with the Halpern iterative scheme for a strongly quasinonexpansive mapping in the setting of a complete geodesic space with curvature bounded above by one. Our result can be applied to the image recovery problem. We also consider the approximation of a fixed point of a nonexpansive mapping and obtain convergence theorems, one of which is a supplement of the result by Pia̧tek with an additional sufficient condition.
- Halpern iteration
- strongly quasinonexpansive
- fixed point
Halpern’s iterative method  is one of the most effective methods to find a fixed point of a nonexpansive mapping, which guarantees strong convergence of the approximating sequence. A remarkable result for nonlinear mappings was obtained by Wittmann  in the setting of Hilbert spaces. Since then, it has been investigated by a large number of researchers, and they have obtained different types of strong convergence theorems for nonexpansive mappings and their variations.
On the other hand, the notion of a strongly nonexpansive mapping was first proposed by Bruck and Reich  as a generalization of firmly nonexpansive mappings. This mapping was later generalized to a strongly quasinonexpansive mapping by Bruck . We propose a new definition of this mapping in the framework of space by imposing a natural bound on the diameter of the space.
The first result of convergence of the Halpern iteration on a complete space was obtained by Saejung  for nonexpansive mappings, and a similar result in the setting of a complete space was proposed by Pia̧tek . The combination of the Halpern iteration and strongly quasinonexpansive mappings was made recently. See [7, 8] and others.
In this paper, we deal with the Halpern iterative scheme for a strongly quasinonexpansive mapping in the setting of a complete space. Then we show that the main result can be applied to the problem of image recovery. We also consider the approximation of a fixed point of a nonexpansive mapping. We point out that the proof of the result by Pia̧tek is not sufficient and we supplement it with an additional sufficient condition for the convergence of the iterative sequence.
Moreover, is said to be Δ-convergent and z is said to be its Δ-limit if z is the unique asymptotic center of any subsequences of .
A geodesic with endpoints is defined as an isometric mapping from the closed segment of real numbers to X whose image connects x and y. If a geodesic exists for every , then X is called a geodesic space.
we call X a space, where is the spherical metric on .
In this paper, we deal with only spaces; however, we remark that all the results can be easily generalized to spaces with positive κ by changing the scale of the space.
For two points x, y in a space X with and , we denote by the point z on a geodesic segment between x and y such that and . A subset C of X is said to be π-convex if belongs to C for every with and .
We refer to  for more details on geodesic spaces including spaces.
where . This simple inequality plays a very important role in this paper.
Let X be a space. Let and suppose that the set of fixed points is not empty. Then T is said to be quasinonexpansive if for every and . T is said to be strongly quasinonexpansive if it is quasinonexpansive, and for every and every sequence in X satisfying that and , it follows that . T is said to be Δ-demiclosed if for any Δ-convergent sequence in X, its Δ-limit belongs to whenever .
The following lemmas are important for our main result.
Lemma 2.1 (Xu )
Let , and be sequences of real numbers such that and for every , , and . Let be a sequence in such that . If for every , then .
Lemma 2.2 (Saejung-Yotkaew )
Let and be sequences of real numbers such that for every . Let be a sequence in such that . Suppose that for every . If for every subsequence of ℕ satisfying , then .
Lemma 2.3 (He-Fang-Lopez-Li )
Let X be a complete space and . If a sequence in X satisfies that and that is Δ-convergent to , then .
As the main theorem of this paper, we prove strong convergence of the iterative sequence to a fixed point of a strongly quasinonexpansive mapping. We adopt the Halpern iterative scheme to generate the sequence. We begin with the following basic lemma, which is one of the main tools for our results.
and hence we obtain the desired result. □
Now, we show the main theorem.
Then converges to .
for all and hence .
for every . Therefore, from the condition (c) we have that .
By Lemma 2.2, we have that , that is, converges to , and we finish the proof. □
In the setting of Hilbert spaces, the image recovery problem can be formulated as to find the nearest point in the intersection of a family of closed convex subsets from a given point by using the metric projection of each subset. In this section, we consider this problem in the setting of complete spaces. As the simplest case, we deal with only two closed convex subsets and such that and generate an iterative sequence converging to the nearest point in from a given point.
and thus , that is, . Hence, is strongly quasinonexpansive.
On the other hand, let be such that and assume that is Δ-convergent to . Then is also Δ-convergent to . Since is a sequence in a closed π-convex subset C, we have that its Δ-limit belongs to C, that is, . It shows that is Δ-demiclosed.
For two strongly quasinonexpansive and Δ-demiclosed mappings having common fixed points, we can create a new strongly quasinonexpansive and Δ-demiclosed mapping whose fixed points are common fixed points of given two mappings. For example, as we have seen above, metric projections to closed and convex sets are strongly quasinonexpansive and Δ-demiclosed. Thus, for given two metric projections to closed convex sets whose intersection is nonempty, the following method is applicable. It is useful to solve the image recovery problem.
Dividing above by , we get the conclusion. □
that is, . Hence, , which means . □
Lemma 4.4 Let X be a space such that for arbitrary v and of X. Let and be mappings from X to X such that . If both and are strongly quasinonexpansive, then so is .
that is, . Then it follows that . Similarly, we have that and . Consequently, we have that and . Hence, we obtain that , which is the desired result. □
Lemma 4.5 Let X be a space such that for arbitrary v and of X. Let and be mappings from X to X such that . If both and are Δ-demiclosed, then so is .
Since is Δ-demiclosed, we have that . In a similar fashion, we have that . Hence , that is, T is Δ-demiclosed. □
Let X be a complete space such that for every , and let and be closed convex subsets of X having the nonempty intersection. Then, for the metric projections and , the mapping is strongly quasinonexpansive and Δ-demiclosed. Moreover, the set of its fixed points is . Applying these facts to Theorem 3.2, we obtain the following result for the image recovery problem for two convex subsets.
Then converges to .
At the end of this paper, we prove two convergence theorems of iterative schemes which approximate a fixed point of a nonexpansive mapping. Firstly, we apply the main result Theorem 3.2 to this problem. We begin with the following lemmas.
Lemma 5.1 A nonexpansive mapping defined on a space is Δ-demiclosed.
a contradiction. Hence, we have that S is Δ-demiclosed. □
for is a strongly quasinonexpansive and Δ-demiclosed mapping such that .
It implies that and hence T is strongly quasinonexpansive.
For the Δ-demiclosedness of T, use Lemmas 4.5 and 5.1 with the fact that the identity mapping is also Δ-demiclosed. □
Applying this lemma and the results in the previous section to Theorem 3.2, we obtain the following convergence theorem of an iterative scheme approximating a fixed point of a nonexpansive mapping.
Then converges to .
The next convergence theorem of an iterative scheme on spaces was first proposed by Pia̧tek . The theorem deals with the Halpern-type iterative sequence. Although the result itself is correct, a part of the proof does not seem to be exact. Precisely, in the proof of the convergence theorem for the explicit iteration process, the author makes use of Xu’s lemma, Lemma 2.1 in this paper. However, the conditions required for this lemma are not verified. We attempt to prove the following theorem as a supplement of the aforementioned result, and moreover, we find another coefficient condition which guarantees convergence of the iterative scheme.
Before showing the result, we need the following lemma which is analogous to [, Lemma 3.3]. The assumption for the length of the edges of the triangle is improved.
which is the desired result. □
Then converges to .
We employ the method used in  for some parts of the proof.
Consequently, we have that by Lemma 2.1. Hence, converges to , which is the desired result. □
The first author is supported by Grant-in-Aid for Scientific Research No. 22540175 from the Japan Society for the Promotion of Science.
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