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Hybrid iterative algorithms for nonexpansive and nonspreading mappings in Hilbert spaces
Fixed Point Theory and Applications volume 2013, Article number: 314 (2013)
Abstract
Recently, Iemoto and Takahashi considered a weak convergence iterative scheme for a nonspreading mapping and a nonexpansive mapping in Hilbert spaces. In this paper, we suggest two hybrid iterative algorithms by modifying Iemoto and Takahashi’s iterative scheme for a countable family of nonspreading mappings and a nonexpansive mapping in Hilbert spaces.
MSC:47H05, 47H09.
1 Introduction and preliminaries
Let H be a Hilbert space and C be a nonempty closed convex subset of H. Let T be a nonlinear mapping of C into itself. We use and to denote the set of fixed points of T and the metric projection from H onto C, respectively.
Recall that T is said to be nonexpansive if
for all .
For approximating the fixed point of a nonexpansive mapping in a Hilbert space, Mann [1] in 1953 introduced the famous iterative scheme as follows:
where T is a nonexpansive mapping of C into itself and is a sequence in . It is well known that defined in (1.2) converges weakly to a fixed point of T.
Attempts to modify the normal Mann iteration method (1.2) for nonexpansive mappings so that strong convergence is guaranteed have recently been made; see, e.g., [2–9].
Let T be a mapping from C into itself. Then T is called nonspreading [3] if
for all . A mapping is called quasi-nonexpansive if and for all and . If T is a nonspreading mapping from C into itself and is nonempty, then T is quasi-nonexpansive. Further, we know that the set of fixed points of each quasi-nonexpansive mapping is closed and convex; see [10].
In [11], by using Moudafi’s iterative scheme [12], Iemoto and Takahashi considered the following weak convergence theorem.
Theorem IT ([11])
Let H be a Hilbert space, and let C be a nonempty closed convex subset of H. Let S be a nonspreading mapping of C into itself, and let T be a nonexpansive mapping of C into itself such that . Define a sequence as follows:
for all , where . Then the following hold:
-
(i)
If and , then converges weakly to ;
-
(ii)
If and , then converges weakly to ;
-
(iii)
If and , then converges weakly to .
In this paper, we modify (1.1) by a hybrid iterative scheme and obtain the strong convergence theorems for a family of nonspreading mappings and a nonexpansive mapping in a Hilbert space.
Let E be a Banach space and K be a nonempty closed convex subset of E. Let be a family of mappings. Then is said to satisfy the AKTT-condition [13] if for each bounded subset B of K, one has
The following is an important result on a family of mappings satisfying the AKTT-condition.
Lemma 1.1 ([13])
Let K be a nonempty and closed subset of a Banach space E, and let be a family of mappings of K into itself which satisfies the AKTT-condition. Then, for each , converges strongly to a point in K. Moreover, let the mapping be defined by
Then, for each bounded subset B of K,
Obviously, if a family of mappings satisfies the AKTT-condition and for each , then it is unnecessary that . To show this, see the following example.
Example 1.1 Let and . Define a family of mappings by
Then satisfy the AKTT-condition. It is easy to see that for each , . Define the mapping by . That is, for all . But .
In this paper, we call that satisfy the AKTT-condition if satisfy the AKTT-condition with .
Lemma 1.2 ([11])
Let C be a nonempty closed subset of a Hilbert space H. Then a mapping is nonspreading if and only if
for all .
By using Lemma 1.2, we get the following simple but important result.
Lemma 1.3 Let H be a Hilbert space and C be a nonempty subset of H. Let be a family of nonspreading mappings of C into itself, and assume that exists for each . Define the mapping by . Then the mapping T is a nonspreading mapping.
Proof In fact, for all , we have
Lemma 1.2 shows that the mapping T is a nonspreading mapping. □
Lemma 1.4 Let C be a closed convex subset of a real Hilbert space H, and let be the metric projection from H onto C (i.e., for , is the only point in C such that ). Given and . Then if and only if the following relation holds:
Lemma 1.5 ([14])
Let H be a real Hilbert space. Then the following equation holds:
2 Main results
Theorem 2.1 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a nonexpansive mapping and be a countable family of nonspreading mappings such that . Let be a sequence generated in the following manner:
where . Assume that is strictly decreasing and . Then the following hold:
-
(i)
If and , then strongly converges to ;
-
(ii)
If and , then converges strongly to .
Proof Obviously, each is closed and convex and hence is closed and convex. Next, we show that for all . To end this, we need to prove that for all . Indeed, for each , we have
This implies that
Therefore, and hence is nonempty for all . On the other hand, from the definition of , we see that for all .
From , we have
Since , one has
This implies that is bounded and hence is bounded.
On the other hand, since for all , we have
for all . From one has
for all . It follows from (2.3) and (2.4) that the limit of exists.
Since and for all and , by Lemma 1.4 one has
It follows from (2.5) that
Since the limit of exists, we get
It follows that is a Cauchy sequence. Since H is a Hilbert space and C is closed and convex, there exists such that
By taking in (2.6), one arrives at
i.e.,
Noticing that , we get
and hence
From (2.7) and (2.9) it follows that
Now we prove (i). Note that
Hence,
On the other hand, for any , from Lemma 1.2 we have
and hence
Note that is strictly decreasing. Hence from (2.11) and (2.12) we get
Since and , from (2.9) and (2.13) it follows that
Since each is a nonspreading mapping, by Lemma 1.2, (2.7) and (2.10), we have
Further, one has
So, we have .
To prove (ii), first we show that . For any , we have
and hence by (2.10) we get
Since , it follows from (2.17) that
From (2.18) and
we get
Since , we get
Now, using (2.19), (2.7) and
which implies that .
Note that (2.9) and (2.19) imply that . Then, repeating (2.11) to (2.16), we get . So, . This completes the proof. □
Theorem 2.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a nonexpansive mapping and be a countable family of nonspreading mappings such that . Let be a sequence generated in the following manner:
where . Assume that satisfies the AKTT-condition. Then the following hold:
-
(i)
If and , then strongly converges to ;
-
(ii)
If and , then converges strongly to .
Proof By a process similar to the proof of Theorem 2.1, we can conclude that converges strongly to some and
We first prove (i). From (2.20) we have
and hence
Since and , we get
Further, by Lemma 1.1 and (2.21), we have
Since each is a nonspreading mapping, Lemma 1.3 shows that T is a nonspreading mapping. Further, by using Lemma 1.2, we have
From (2.21) and (2.23) it follows that
It follows that . Since satisfies the AKTT-condition, one has . This completes (i).
Next we show (ii). By a process similar to the proof of Theorem 2.1 and from (2.22) to (2.24), we can get that
Finally, by
and
we get . Since satisfies the AKTT-condition, we conclude that . This completes (ii). □
Letting for all in Theorem 2.1 and Theorem 2.2, we get the following corollary.
Corollary 2.1 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a nonexpansive mapping and be a nonspreading mapping such that . Let be a sequence generated in the following manner:
where . Then the following hold:
-
(i)
If and , then strongly converges to ;
-
(ii)
If and , then converges strongly to with .
Letting in Theorems 2.1 and 2.2, we get the following corollary.
Corollary 2.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a countable family of nonspreading mappings such that . Let be a sequence generated in the following manner:
where . Assume that is strictly decreasing and . Then if , then strongly converges to .
Corollary 2.3 Let C be a nonempty closed convex subset of a Hilbert space H. Let be a countable family of nonspreading mappings such that . Let be a sequence generated in the following manner:
where . Assume that satisfies the AKTT-condition. Then if , then strongly converges to .
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Acknowledgements
This work is supported by the Fundamental Research Funds for the Central Universities (Grant Number: 13MS109).
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Wang, S. Hybrid iterative algorithms for nonexpansive and nonspreading mappings in Hilbert spaces. Fixed Point Theory Appl 2013, 314 (2013). https://doi.org/10.1186/1687-1812-2013-314
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DOI: https://doi.org/10.1186/1687-1812-2013-314