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Strong convergence of an iterative process for a family of strictly pseudocontractive mappings
Fixed Point Theory and Applications volume 2013, Article number: 117 (2013)
Abstract
In this article, fixed point problems of a family of strictly pseudocontractive mappings are investigated based on an iterative process. Strong convergence of the iterative process is obtained in a real 2-uniformly Banach space.
MSC:47H09, 47J05, 47J25.
1 Introduction and preliminaries
Throughout this paper, we always assume that E is a real Banach space. Let be the dual space of E. Let () denote the generalized duality mapping from E into given by
where denotes the generalized duality pairing. In particular, is called the normalized duality mapping, which is usually denoted by J. In this paper, we use j to denote the single-valued normalized duality mapping. It is known that if . If E is a Hilbert space, then , the identity mapping. Further, we have the following properties of the generalized duality mapping :
-
(1)
for all and ;
-
(2)
for all .
A Banach space E is said to be smooth if the limit
exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for all . The norm of E is said to be Fréchet differentiable if, for any , the above limit is attained uniformly for all . The modulus of smoothness of E is the function defined by
The Banach space E is uniformly smooth if and only if . Let . The Banach space E is said to be q-uniformly smooth if there exists a constant such that . It is shown in [1] that there is no Banach space which is q-uniformly smooth with . Hilbert spaces, (or ) spaces and Sobolev space , where , are 2-uniformly smooth.
Let C be a nonempty closed convex subset of E and let be a mapping. In this paper, we use to denote the fixed point set of T. A mapping T is said to be κ-contractive iff there exists a constant such that
A mapping T is said to be nonexpansive iff
A mapping T is said to be κ-strictly pseudocontractive iff there exist a constant and such that
It is clear that (1.1) is equivalent to the following:
The class of κ-strictly pseudocontractive mappings was first introduced by Browder and Petryshyn [2] in Hilbert spaces. A mapping T is said to be pseudocontractive iff there exists such that
A mapping T is said to be κ-strongly pseudocontractive iff there exist a constant and such that
In 1974, Deimling [3] proved the existence of fixed points of continuous κ-strongly pseudocontractive mappings in Banach spaces; see [3] for more details. We remark that the class of κ-strongly pseudocontractive mappings is independent of the class of κ-strictly pseudocontractive mappings. This can be seen from Zhou [4]. Lipschitz pseudocontractive mappings may not be κ-strictly pseudocontractive mappings, which can be seen from Chidume and Mutangadura [5].
One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping; for more details, see [6–12] and the references therein. More precisely, take and define a contraction by
where is a fixed point. Banach’s contraction mapping principle guarantees that has a unique fixed point in C. In the case of T having a fixed point, Browder [7] proved that converges strongly to a fixed point of T in the framework of Hilbert spaces. Reich [10] extended Browder’s result to the setting of Banach spaces and proved that if E is a uniformly smooth Banach space, then converges strongly to a fixed point of T and the limit defines the (unique) sunny nonexpansive retraction from C onto ; see [10] for more details.
Recall that the normal Mann iterative process was introduced by Mann [13] in 1953. Recently, the construction of fixed points for nonexpansive mappings via the normal Mann iterative process has been extensively investigated by many authors. The normal Mann iterative process generates a sequence in the following manner:
where the sequence is in the interval .
In an infinite-dimensional Hilbert space, the normal Mann iteration algorithm has only weak convergence; see [14] for more details. In many disciplines, including economics [15], image recovery [16] and control theory [17], problems arise in infinite dimension spaces. In such problems, strong convergence is often much more desirable than weak convergence for it translates the physically tangible property so that the energy of the error between the iterate and the solution x eventually becomes arbitrarily small.
Recently, many authors have tried to modify the normal Mann iteration process to have strong convergence for nonexpansive mappings and κ-strictly pseudocontractive mappings; see [18–36] and the references therein.
Let D be a nonempty subset of C. Let . Q is said to be a contraction iff ; sunny iff for each and , we have ; sunny nonexpansive retraction iff Q sunny, nonexpansive and contraction. K is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D. The following result, which was established in [37] and [38], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Let be a retraction, and let j be the normalized duality mapping on E. Then the following are equivalent:
-
(1)
Q is sunny and nonexpansive;
-
(2)
, ;
-
(3)
, , .
In this paper, we investigate the problem of modifying the normal Mann iteration process for a family of κ-strictly pseudocontractive mappings. Strong convergence of the purposed iterative process is obtained in a real 2-uniformly Banach space. In order to prove our main results, we need the following tools.
Lemma 1.1 [1]
Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
Lemma 1.2 [31]
Let C be a nonempty subset of a real 2-uniformly smooth Banach space E and let be a κ-strict pseudocontraction. For , we define for every . Then, as , is nonexpansive such that .
Lemma 1.3 [39]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(i)
;
-
(ii)
or .
Then .
Lemma 1.4 [40]
Let E be a real smooth Banach space. Then the following inequality holds:
Lemma 1.5 [29]
Let E be a smooth Banach space and let C be a nonempty convex subset of E. Given an integer , assume that is a finite family of -strict pseudocontractions such that . Assume that is a positive sequence such that . Then .
Lemma 1.6 [10]
Let E be a real uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let be a nonexpansive mapping with a fixed point and let be a contraction. For each , let be the unique solution of the equation . Then converges to a fixed point of T as and defines the unique sunny nonexpansive retraction from C onto .
2 Main results
Theorem 2.1 Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E with the best smooth constant K and let N be some positive integer. Let be a -strictly pseudocontractive mapping for each . Assume that . Let f be an α-contractive mapping. Let be a sequence generated in the following process:
where , and are real number sequences in satisfying the following restrictions:
-
(a)
, , ,
-
(b)
, ,
-
(c)
,
where β is some real number, and . Then converges strongly as to some point in , which is the unique solution in to the following variational inequality:
Proof The proof is split into four steps.
Step 1. Show that and are bounded.
Putting , we see that T is a κ-strictly pseudocontractive mapping. Indeed, we have the following:
This proves that T is a κ-strictly pseudocontractive mapping. Fix . It follows from Lemma 1.1 that
This implies that
This in turn implies that
which gives that the sequence is bounded, so is . This completes step 1.
Step 2. Show that as .
Put , . It follows from Lemma 1.2 that
Notice that
It follows from (2.2) that
In view of Lemma 1.3, we obtain from the restrictions (a) and (b) that
Notice that
In view of the restriction (a), we obtain that . On the other hand, we have . This in turn implies that . It follows from the restriction (b) that
This completes step 2.
Step 3. Show that
where , where Q is a sunny nonexpansive retraction from C onto , is the strong limit of the sequence defined by
It follows that
For any , we see from Lemma 1.4 that
where
It follows from (2.6) that
This implies that
Since E is 2-uniformly smooth, is uniformly continuous on any bounded sets of E, which ensures that the and are interchangeable, and hence
This shows that (2.5) holds. This completes the proof of step 3.
Step 4. Show that as .
It follows from (2.1) that . In view of Lemma 1.4, we see that
It then follows that
It follows from the restrictions (a) and (b) that
and
This implies from Lemma 1.3 that as . This completes the proof. □
For a single mapping, we have the following.
Corollary 2.2 Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E with the best smooth constant K. Let be a κ-strictly pseudocontractive mapping such that . Let f be an α-contractive mapping. Let be a sequence generated in the following process:
where , and are real number sequences in satisfying the following restrictions:
-
(a)
, , ;
-
(b)
, ,
where β is some real number, and . Then converges strongly as to some point in , which is the unique solution in , to the following variational inequality:
If E is a Hilbert space, then the best smooth constant . The following result can be deduced from Theorem 2.1 immediately.
Corollary 2.3 Let C be a nonempty closed convex subset of a real Hilbert space E and let N be some positive integer. Let be a -strictly pseudocontractive mapping for each . Assume that . Let f be an α-contractive mapping. Let be a sequence generated in the following process:
where , and are real number sequences in satisfying the following restrictions:
-
(a)
, , ;
-
(b)
, ;
-
(c)
,
where β is some real number, and . Then converges strongly as to some point in , which is the unique solution in , to the following variational inequality:
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Acknowledgements
This research was supported by the Natural Science Foundation of Hebei Province (A2010001943), the Science Foundation of Shijiazhuang Science and Technology Bureau (121130971) and the Science Foundation of Beijing Jiaotong University (2011YJS075). The authors are grateful to the referees for their valuable comments and suggestions which improved the contents of the article.
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Qing, Y., Cho, S.Y. & Shang, M. Strong convergence of an iterative process for a family of strictly pseudocontractive mappings. Fixed Point Theory Appl 2013, 117 (2013). https://doi.org/10.1186/1687-1812-2013-117
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DOI: https://doi.org/10.1186/1687-1812-2013-117