Strong convergence of an iterative process for a family of strictly pseudocontractive mappings
© Qing et al.; licensee Springer 2013
Received: 13 February 2013
Accepted: 19 April 2013
Published: 2 May 2013
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
for all and ;
for all .
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  that there is no Banach space which is q-uniformly smooth with . Hilbert spaces, (or ) spaces and Sobolev space , where , are 2-uniformly smooth.
In 1974, Deimling  proved the existence of fixed points of continuous κ-strongly pseudocontractive mappings in Banach spaces; see  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 . Lipschitz pseudocontractive mappings may not be κ-strictly pseudocontractive mappings, which can be seen from Chidume and Mutangadura .
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  proved that converges strongly to a fixed point of T in the framework of Hilbert spaces. Reich  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  for more details.
where the sequence is in the interval .
In an infinite-dimensional Hilbert space, the normal Mann iteration algorithm has only weak convergence; see  for more details. In many disciplines, including economics , image recovery  and control theory , 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  and , describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Q is sunny and nonexpansive;
, , .
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 
Lemma 1.2 
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 
Lemma 1.4 
Lemma 1.5 
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 
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
, , ,
Proof The proof is split into four steps.
Step 1. Show that and are bounded.
which gives that the sequence is bounded, so is . This completes step 1.
Step 2. Show that as .
This completes step 2.
This shows that (2.5) holds. This completes the proof of step 3.
Step 4. Show that as .
This implies from Lemma 1.3 that as . This completes the proof. □
For a single mapping, we have the following.
, , ;
If E is a Hilbert space, then the best smooth constant . The following result can be deduced from Theorem 2.1 immediately.
, , ;
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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