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General alternative regularization methods for nonexpansive mappings in Hilbert spaces
Fixed Point Theory and Applications volume 2014, Article number: 203 (2014)
Abstract
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm . Let be a nonexpansive mapping with a nonempty set of fixed points and let be a Lipschitzian strong pseudo-contraction. We first point out that the sequence generated by the usual viscosity approximation method may not converge to a fixed point of T, even not bounded. Secondly, we prove that if the sequence satisfies the conditions: (i) , (ii) and (iii) or , then the sequence generated by a general alternative regularization method: converges strongly to a fixed point of T, which also solves the variational inequality problem: finding an element such that for all . Furthermore, we prove that if T is replaced with the sequence of average mappings () such that , where and are two positive constants, then the same convergence result holds provided conditions (i) and (ii) are satisfied. Finally, an algorithm for finding a common fixed point of a family of finite nonexpansive mappings is also proposed and its strong convergence is proved. Our results in this paper extend and improve the alternative regularization methods proposed by HK Xu.
MSC:47H09, 47H10, 65K10.
1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm and let be a α-contractive mapping, i.e., there exists a constant such that holds for all . Let be a nonexpansive mapping, i.e., for all . Throughout this article, the set of fixed points of T, indicated by , 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 [1]) that is nonexpansive and a characteristic inequality holds: given and , then if and only if
Since is a closed convex subset of H, so the metric projection is valid.
Recall that a mapping is said to be L-Lipschitzian, if there exists a positive constant L such that
and a mapping is said to be α-strongly pseudo-contractive, if there exists a constant such that
In this case, we also call h a α-strong pseudo-contraction.
It is very easy to see that a α-contractive mapping is a α-strongly pseudo-contractive and α-Lipschitzian mapping, i.e., the class of contractive mappings is a proper subset of the class of Lipschitzian strong pseudo-contractions. The class of Lipschitzian strong pseudo-contractions will be used repeatedly in the sequel.
Recall that a mapping is said to be η-strongly monotone, if there exists a positive constant η such that
The variational inequality problem [2] can mathematically be formulated as the problem of finding a point with the property
where K is a nonempty closed convex subset of H and is a nonlinear operator. It is well known that [3] if is a Lipschitzian and strongly monotone operator, then the variational inequality problem has a unique solution.
Many iteration processes are often used to approximate a fixed point of a nonexpansive mapping in a Hilbert space or a Banach space (for example, see [1] and [4–20]). One of them is now known as Halpern’s iteration process [4] 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 in the setting of Hilbert spaces is as follows.
Theorem 1.1 ([4])
If satisfies the conditions:
-
(i)
();
-
(ii)
;
-
(iii)
or ;
then the sequence generated by (1.1) converges strongly to a fixed point of T such that , that is,
Xu [5] proposed an alternative regularization method:
and studied its strong convergence in the setting of Hilbert spaces and Banach spaces, respectively. Indeed, in the setting of Hilbert spaces, one can proved that for algorithm (1.2) the same convergence result as that of Theorem 1.1 holds under conditions (i)-(iii) above.
As an extension to Halpern’s iteration process, Moudafi proposed [15] the viscosity approximation method: take an initial guess arbitrarily and define recursively by
where is a sequence in the interval . Moudafi proved the following result in Hilbert spaces.
Theorem 1.2 ([15])
If satisfies the same conditions (i)-(iii) as above, then the sequence generated by (1.3) converges strongly to a fixed point of T, which also solves the variational inequality problem: finding an element such that
Xu studied the viscosity approximation method in the setting of Banach spaces and obtained the strong convergence theorems [16].
Similar to algorithm (1.2), we can naturally consider a general alternative regularization method: take an initial guess arbitrarily and define recursively by
where is a sequence in the interval . In fact, in the setting of Hilbert spaces, it is not difficult to prove by an argument very similar to the proof of Theorem 1.2 that for algorithm (1.4) the same result as that of Theorem 1.2 holds under conditions (i)-(iii) above.
The main purpose of this paper is to consider a very natural question: if algorithms (1.3) and (1.4) can be extended to more general cases, more precisely, can we replace a contractive mapping f with a Lipschitzian strong pseudo-contraction h so that the same convergence result as that of Theorem 1.2 is still guaranteed under conditions (i)-(iii) as above? The answer to this question is negative for algorithm (1.3) unfortunately but is sure for algorithm (1.4) fortunately. In this sense, it seems reasonable to deem that algorithm (1.4) is better that algorithm (1.3).
Now we illustrate a fact via a very simple example that if f in algorithm (1.3) is replaced with a Lipschitzian and strongly pseudo-contractive mapping h, then strong convergence (even boundedness) of the iteration sequence may not be guaranteed, in general. Indeed, take , , and defined by
Noting that T is just a rotation operator over , we see that T is a nonexpansive mapping. Moreover, noting the fact that
holds for all , it is easy to see that for any positive constant κ and any , is a κ-Lipschitzian and α-strongly pseudo-contractive mapping.
If f in algorithm (1.3) is replaced with (i.e., ), then (1.3) is of the form:
Since T is a rotation operator over , the sequence generated by (1.5) does not converge to , the unique fixed point of T, unless we take the initial guess . On the other hand, if f in algorithm (1.3) is replaced with (i.e., ), thus (1.3) is rendered in the form
Consequently, noting that , , we have
Since , holds and thus this implies that the sequence generated by (1.6) is not bounded provided .
The rest of this paper is organized as follows. In order to prove our main results, some useful facts and tools are listed as lemmas in the next section. In Section 3, we prove that if a contractive mapping f in algorithm (1.4) is replaced with a Lipschitzian strong pseudo-contraction h, then the same convergence result as that of Theorem 1.2 is still guaranteed under conditions (i)-(iii) as above. Furthermore, we prove that if T is replaced with the sequence of average mappings () such that , where and are two positive constants, then the same result still holds provided conditions (i) and (ii) are satisfied. In the last section, an algorithm for finding a common fixed point of a family of finite nonexpansive mappings is also proposed and its strong convergence is proved.
We will use the notations:
-
1.
⇀ for weak convergence and → for strong convergence.
-
2.
denotes the weak ω-limit set of .
-
3.
means that B is the definition of A.
2 Preliminaries
We need some facts and tools in a real Hilbert space H, which are listed as lemmas below.
Lemma 2.1 The following relation holds in a real Hilbert space H:
Assume is a sequence of nonnegative real numbers satisfying the property
If , and satisfy the conditions:
-
(i)
,
-
(ii)
,
-
(iii)
,
then .
Lemma 2.3 ([22])
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 2.4 For each , the following identity holds in a real Hilbert space H:
Lemma 2.5 ([23])
Assume is a sequence of nonnegative real numbers such that
where is a sequence in , is a sequence of nonnegative real numbers and and are two sequences in ℝ such that
-
(i)
,
-
(ii)
,
-
(iii)
implies for any subsequence .
Then .
3 Algorithms for one mapping
Throughout this section, we will assume that H is a real Hilbert space with the inner product and the norm , C is a closed convex subset of H and is a L-Lipschitzian and α-strongly pseudo-contractive mapping, i.e., there exist positive constants L and such that
It is obvious that if h is a α-strong pseudo-contraction, then is a -strongly monotone mapping, i.e.,
where I denotes the identity operator.
Our first result is as follows.
Theorem 3.1 Let be a L-Lipschitzian and α-strongly pseudo-contractive mapping and let be a nonexpansive mapping. If the sequence satisfies the conditions:
-
(i)
();
-
(ii)
;
-
(iii)
or ;
then the sequence generated by the algorithm
where is selected in C arbitrarily, converges strongly to a fixed point of T, which also solves the variational inequality problem: finding an element such that
Proof Noting that is a -Lipschitzian and -strongly monotone mapping, so the variational inequality problem (3.4) has a unique solution, which is denoted by . Now we try to prove that .
Firstly, we deduce from (3.1)-(3.3) that
Obviously
where β is a positive constant such that . Thus, the combination of (3.5) and (3.6) leads to
Noting the fact that and
we assert that there exists some integer such that and
hold for all . So we see from (3.7) and (3.8) that for all , the following relation holds:
Consequently
and inductively
This means that is bounded, so is .
From (3.3), we have
due to the boundedness of and . Now we show that . Setting , , we derive from Lemma 2.1, (3.1) and (3.2) that
and
Noting the boundedness of and , it follows from (3.11)-(3.13) that
where and M is a positive constant independent on n. Observing that
we get from conditions (i) and (ii) that and hold. Hence, this together with condition (iii) allows us to assert by using Lemma 2.2. From this together with (3.10) one concludes that and hence we obtain by using Lemma 2.3.
Finally, we prove (). Again using (3.5), we also have
where and
Take a subsequence such that
Without loss of the generality, we assume that there exists some (noting that holds) such that (otherwise, we may select some subsequence of with this property). Hence, we have
noting that is the unique solution of the variational inequality (3.4). Consequently, it is easy to verify that by using (3.15). This together with and allows us to use Lemma 2.2 to (3.14) to obtain
□
Theorem 3.2 Let be a L-Lipschitzian and α-strongly pseudo-contractive mapping and let be a nonexpansive mapping. Let , where . Take arbitrarily and define a sequence by the process
where . If and satisfy the conditions:
-
(i)
();
-
(ii)
;
-
(iii)
there exist two constants and such that for all ,
then the sequence converges strongly to a fixed point of T, which also solves the variational inequality problem: finding a point such that
Proof Firstly, noting that
we assert that is bounded by an argument similar to the proof of Theorem 3.1. Moreover, in a way similar to getting (3.5), we have from (3.17)
Secondly, setting and using Lemma 2.4, we have from (3.16) and (3.18)
Setting
thus (3.18) and (3.19) can be rewritten as the form
Since , () and hold obviously, so in order to complete the proof by using Lemma 2.5, it suffices to verify that () implies that
for any subsequence .
Indeed, () implies that due to condition (iii). Furthermore, the relation
together with the fact that
allows us to get (). Using Lemma 2.3, we get . Thus we have
and hence holds. □
4 Algorithm for several mappings
In this section, we turn to considering an algorithm for finding a common fixed point of a family of finite nonexpansive mappings.
Let H be a real Hilbert space and let C be a closed convex subset of H. Let N be an integer such that and let () be a family of finite nonexpansive mappings. Set
where for all and .
Our main result in this section is as follows.
Theorem 4.1 Let be a L-Lipschitzian and α-strongly pseudo-contractive mapping. Take a initial guess arbitrarily and define a sequence by
where and is given as above. If and satisfy the conditions:
-
(i)
();
-
(ii)
;
-
(iii)
there exist two constants and such that for all and ,
then the sequence generated by (4.1) converges strongly to a common fixed point of , which also solves the variational inequality problem: finding an element such that
Proof Without loss of the generality, we only give the proof for the case where .
It is clear that the variational inequality (4.2) has a unique solution, which is denoted by in the sequel. An argument very similar to the proof of Theorem 3.2 allows us to verify that is bounded (so is ) and the following relation holds:
On the other hand, setting , we have, by using Lemma 2.4,
Similar to (3.18), it is easy to verify that
Combining (4.4) and (4.5), we derive that
Setting
thus (4.3) and (4.6) can be rewritten in the following forms, respectively:
By using Lemma 2.5, in order to complete the proof, it suffices to show that () implies that
In fact, noting condition (iii), we get from , and all hold. Consequently, follows from the inequality
and the fact that () and hence holds. On the other hand, using , and the relation
we conclude that and this means that also holds. Thus we have proved that
Moreover, we have
and hence
□
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant no. 11201476) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.
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Yang, C., He, S. General alternative regularization methods for nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl 2014, 203 (2014). https://doi.org/10.1186/1687-1812-2014-203
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DOI: https://doi.org/10.1186/1687-1812-2014-203
Keywords
- fixed point
- nonexpansive mapping
- strong pseudo-contraction
- viscosity approximation method
- general alternative regularization method