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Another type of Mann iterative scheme for two mappings in a complete geodesic space
Journal of Inequalities and Applications volume 2014, Article number: 72 (2014)
Abstract
In this paper, we show a Δ-convergence theorem for a Mann iteration procedure in a complete geodesic space with two quasinonexpansive and Δ-demiclosed mappings. The proposed method is different from known procedures with respect to the order of taking the convex combination.
1 Introduction
The fixed point approximation has been studied in a variety of ways and its results are useful for the other studies. In 1953, Mann [1] introduced an iteration procedure for approximating fixed points of a nonexpansive mapping T in a Hilbert space. Later, Reich [2] discussed this iteration procedure in a uniformly convex Banach space whose norm is Fréchet differentiable. In 1998, Takahashi and Tamura [3] considered an iteration procedure with two nonexpansive mappings and obtained weak convergence theorems for this procedure in a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. On the other hand, in 2008, Dhompongsa and Panyanak [4] proved the following theorem.
Theorem 1.1 Let C be a bounded closed convex subset of a complete space and a nonexpansive mapping. For any initial point in C, define the Mann iterative sequence by
where is a sequence in , with the restrictions that diverges and . Then Δ-converges to a fixed point of T.
Further, in a space, Kimura et al. [5] proved the Δ-convergence theorem for a family of nonexpansive mappings including the following scheme:
In a Hilbert space H, the following equality holds for any :
where such that and . However, in spaces with , it does not hold in general, that is, the value of the convex combination taken twice depends on their order. Thus, the following formulas are different in general:
In this paper, we show an analogous result to Theorem 1.1 using the iterative scheme (1) in a complete space with two quasinonexpansive and Δ-demiclosed mappings. We also deal with the image recovery problem for two closed convex sets.
2 Preliminaries
Let X be a metric space. For , a mapping is said to be a geodesic if c satisfies , and for all . An image of c is called a geodesic segment joining x and y. For , X is said to be an r-geodesic metric space if, for any with , there exists a geodesic segment . In particular, if a segment is unique for any with , then X is said to be a uniquely r-geodesic metric space. In what follows, we always assume for any . Thus, we say X is a geodesic metric space instead of a -geodesic metric space. For the more general case, see [6].
Let X be a uniquely geodesic metric space. A geodesic triangle is defined by . Let M be the two-dimensional unit sphere in . For , a triangle is called a comparison triangle of if , , . Further, for any and , if satisfies and , then z is denoted by . A point is called a comparison point of if . X is said to be a space if, for any and its comparison points , the inequality holds.
Let X be a geodesic metric space and a bounded sequence of X. For , we put . The asymptotic radius of is defined by . Further, the asymptotic center of is defined by . If, for any subsequences of , , i.e., their asymptotic center consists of the unique element , then we say Δ-converges to and we denote it by .
Let X be a metric space. A mapping is said to be a nonexpansive if T satisfies for any . The set of fixed points of T is denoted by . Further, a mapping with is said to be a quasinonexpansive if T satisfies for any and . Moreover, T is said to be Δ-demiclosed if, for any bounded sequence and satisfying and , we have .
3 Tools for the main results
In this section, we introduce some tools for using the main theorem.
Theorem 3.1 (Kimura and Satô [7])
Let be a geodesic triangle in a space such that . Let for some . Then
Corollary 3.2 (Kimura and Satô [8])
Let be a geodesic triangle in a space such that . Let for some . Then
Theorem 3.3 (Espínola and Fernández-León [9])
Let X be a complete space and a sequence in X. If , then the following hold.
-
(i)
consists of exactly one point;
-
(ii)
has a Δ-convergent subsequence.
Theorem 3.4 (Kimura and Satô [8])
Let X be a metric space and T a mapping from X into itself. If T is a nonexpansive with , then T is quasinonexpansive and Δ-demiclosed.
The following lemmas are important properties of real numbers and they are easy to show. Thus, we omit the proofs.
Lemma 3.5 Let δ be a real number such that and , real sequences satisfying , and . Then .
Lemma 3.6 Let and , bounded real sequences satisfying , and . Then .
Lemma 3.7 Let and be bounded real sequences satisfying . Then .
4 The main result
In this section, we show the main result.
Theorem 4.1 Let X be a complete space such that for any , . Let S and T be quasinonexpansive and Δ-demiclosed mappings from X into itself with . Let , and be sequences of . Define a sequence by the following recurrence formula: and
for . Then Δ-converges to a common fixed point of S and T.
Proof Let . By Corollary 3.2, we have
Then, by Corollary 3.2 again, we have
So, we get for all and there exists .
Furthermore, by Theorem 3.1, we have
and
Let , and for . If there exists such that , then we have and since
and the proof is finished. So, we may assume or for all .
If and , then we have . From (2), (3), and Corollary 3.2, we get
Dividing by , we get
If and , then we have . In a similar way as above, we get
If and , then from (2), (3), and Corollary 3.2, we get
Dividing by , we get
Therefore, (4) and (5) can be reduced to the inequality (6) and it is equivalent to
where for . It follows that . Since for , we get
Then we show that there exists such that for all , the following hold:
and
First, we show the right inequality of (8). Since for , we get . Hence we get
By the same method as above, the right inequality of (9) also holds. Next, let us show the left inequality of (8). If it does not hold, then letting
we can find a subsequence such that for and . Since and , we have is bounded. Therefore, by taking a subsequence again if necessary, we may assume that converges to . Then, by (7), we get . Hence we may assume that for all . Since , , , and , we also have
Let ρ be a real number such that
Then, by (10), we get
Then, by (11) and (12), we have
Thus, as , we have
This is a contradiction. We also obtain the left inequality of (9) in a similar way. Hence we get
by Lemma 3.5, (8), and (9). Furthermore, from (14), we get
By Lemma 3.6 and (15), we get
Moreover, by Lemma 3.7 and (16), we get
Hence we get
and we have
Since for , we have and thus, . Therefore, we get and we also get . It implies and .
Next, let be a subsequence of . Since , by Theorem 3.3(i), there exists a unique asymptotic center of . Moreover, since , by Theorem 3.3(ii), there exists a subsequence of such that . Further, since , and S, T are Δ-demiclosed, we have . Then we can show that , i.e., . If not, from the uniqueness of the asymptotic centers , of , , respectively, due to Theorem 3.3(i), we have
This is a contradiction. Hence we get . Next, we show that for any subsequences of , their asymptotic center consists of the unique element. Let , be subsequences of , and . We show by using contradiction. Assume . Then and by Theorem 3.3(i). It follows that
This is a contradiction. Hence we get . Therefore, we have Δ-converges to a common fixed point of S and T. □
By Theorem 3.4, we know that a nonexpansive mapping having a fixed point satisfies the assumptions in Theorem 4.1. Thus, we get the following result.
Corollary 4.2 Let X be a complete space such that for any , . Let S and T be nonexpansive mappings of X into itself such that . Let , and be sequences in . Define a sequence as the following recurrence formula: and
for . Then Δ-converges to a common fixed point of S and T.
5 An application to the image recovery
The image recovery problem is formulated as to find the nearest point in the intersection of family of closed convex subsets from a given point by using corresponding metric projection of each subset. In this section, we consider this problem for two subsets of a complete space.
Theorem 5.1 Let X be a complete space such that for any , . Let and be nonempty closed convex subsets of X such that . Let and be metric projections onto and , respectively. Let , and be real sequences in . Define a sequence by the following recurrence formula: and
for . Then Δ-converges to a fixed point of the intersection of and .
Proof We see that and are quasinonexpansive [9] and Δ-demiclosed [8]. Further, we also get and . Thus, letting and in Theorem 4.1, we obtain the desired result. □
References
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Acknowledgements
The authors thank the anonymous referees for their valuable comments and suggestions. 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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Kimura, Y., Nakagawa, K. Another type of Mann iterative scheme for two mappings in a complete geodesic space. J Inequal Appl 2014, 72 (2014). https://doi.org/10.1186/1029-242X-2014-72
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DOI: https://doi.org/10.1186/1029-242X-2014-72