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A hybrid iteration for asymptotically strictly pseudocontractive mappings
Journal of Inequalities and Applications volume 2014, Article number: 374 (2014)
In this paper, we propose a new hybrid iteration for a finite family of asymptotically strictly pseudocontractive mappings. We also prove that such a sequence converges strongly to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings. Results in the paper extend and improve recent results in the literature.
Let H be a real Hilbert space, C be a nonempty closed convex subset of H. A mapping is called Lipschitz or Lipschitz continuous if there exists such that
If , then T is called nonexpansive, and if , then T is called a contraction. It follows from (1.1) that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz.
A mapping is said to be a λ-strictly pseudocontractive mapping in the sense of Browder-Petryshyn  if there exists a constant such that
If , then T is said to be a pseudocontractive mapping, i.e.,
The class of strictly pseudocontractive mappings falls into the one between the class of nonexpansive mappings and that of pseudocontractive mappings. The class of strict pseudocontractive mappings has more powerful applications than nonexpansive mappings, see Scherzer .
A mapping is said to be an asymptotically λ-strictly pseudocontractive mapping  if there exists a sequence with , and a constant such that
We now give an example to show that a λ-strictly asymptotically pseudocontractive mapping is not necessarily a λ-strictly pseudocontractive mapping.
Example 1.1 Consider , and let . It is clear that is a normed linear space with respect to the norm
Define by the rule
where is a real sequence satisfying , , and . By definition
for all , . Since , it follows that T is an asymptotically strictly pseudocontractive mapping.
We now show that the T is not a λ-strictly pseudocontractive mapping. Choose , and . Then
Hence T is not a λ-strictly pseudocontractive mapping.
The study on iterative methods for strict pseudocontractive mappings was initiated by Browder and Petryshyn  in 1967, but the iterative methods for strict pseudocontractive mappings are far less developed than those for nonexpansive mappings. The probable reason is the second term appearing on the right-hand side of (1.2), which impedes the convergence analysis. Therefore it is interesting to develop the iteration methods for strict pseudocontractive mappings.
Browder and Petryshyn  showed that if a λ-strict pseudocontractive mapping T has a fixed point in C, then starting with an initial , the sequence generated by the formula
where α is a constant such that , converges weakly to a fixed point of T.
where is a sequence in . Iteration (1.5) is called the Mann iteration .
Another interesting problem is to find a common fixed point of a finite family of strict pseudocontractive mappings. One approach to study the problem is cyclic algorithm, in which sequence is generated cyclically by
where with , .
However, the convergence of both algorithms (1.5) and (1.6) can only be weak in an infinite dimensional space. So, in order to have strong convergence, one must modify these algorithms.
One such modification of Mann’s algorithm for nonexpansive mappings is given by Nakajo and Takahashi , in which a modified algorithm is obtained by applying additional projections onto the intersection of two half-spaces and is guaranteed to have strong convergence. The sequence is produced as follows:
here is the metric projection of H onto C and , are given by
Marino and Xu  proposed the following modification for strict pseudocontractive mappings in which the sequence is given by the same formula (1.7) with given by
where and are given by formulas (1.9) and (1.10), respectively.
Thakur  extended the idea of Marino and Xu  to asymptotically strict pseudocontractive mappings. Recently, Yao and Chen  proposed a new hybrid method for strict pseudocontractive mappings, in which , , are given by the same formulas (1.7), (1.8), (1.10) respectively, and
where , .
Takahashi et al.  introduced the idea of shrinking projection method for nonexpansive mappings, in which projection is applied on a single set. Here the sequence is produced by the formula
where is given by
and is given by the same formula (1.9).
Inchan and Nammanee  modified the shrinking projection method for asymptotically strict pseudocontractive mappings, in which the sequence is generated by the same formula (1.12) and
Motivated and inspired by the studies going on in this direction, we now propose the modified shrinking projection method for a finite family of asymptotically λ-strict pseudocontractive mappings.
Let C be a bounded closed convex subset of a Hilbert space H and be a finite family of asymptotically -strict pseudocontractive mappings with Lipschitz constant , , and for all such that . Set and , . For arbitrarily chosen , let and , define a sequence as
where is some constant and
also for each , it can be written as , where and is a positive integer with as .
We shall prove that the iteration generated by (1.14) converges strongly to .
This section collects some lemmas which will be used in the proofs for the main results in the next section.
We will use the following notation:
⇀ for weak convergence and → for strong convergence.
denotes the weak ω-limit set of .
the set of fixed points of T.
Lemma 2.1 ()
The following identities hold in a Hilbert space H:
Lemma 2.2 ()
Assume that C is a closed and convex subset of a Hilbert space H, and let be an asymptotically λ-strict pseudocontraction with . Then:
For each , satisfies the Lipschitz condition
for all , where .
If is a sequence in C with the properties and , then , i.e., is demiclosed at 0.
The fixed point set of T is closed and convex so that the projection is well defined.
Lemma 2.3 ()
Let H be a real Hilbert space. Given a closed convex subset and points . Given also a real number a. The set
is convex (and closed).
Lemma 2.4 Let C be a closed convex subset of a real Hilbert space H. Given and . Then if and only if there holds the relation
where is the nearest point projection from H onto C.
3 Main results
In this section, we prove a strong convergence theorem by the hybrid method for a finite family of asymptotically -strictly pseudocontractive mappings in Hilbert spaces.
Theorem 3.1 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H. Let be a finite family of asymptotically -strictly pseudocontractive mappings of C into itself for some with Lipschitz constant , and for all such that and let . For and , assume that the control sequence is chosen such that . Then generated by (1.14) converges strongly to .
Proof Suppose , and . By Lemma 2.3, set is closed and convex.
Now, for every , we prove that and is well defined.
We use the method of mathematical induction. For any , we have . Hence . Now assume that for some . Then, for any , we have
It follows that and . Hence for all . Since is closed and convex for all , this implies that is well defined.
Now, we prove that is bounded.
Since , then by Lemma 2.4 we have
As , we have
So, for , we have
This implies that
Hence is bounded.
From and , by Lemma 2.4 we have
So, for , we have, for ,
This implies that
Next, we show that exists.
Using (3.1), we have
Since exists, it follows that
Hence is a Cauchy sequence, and so convergent.
Since , we have
By the definition of , we have
Since , by (1.14) we have
which implies that
Using (3.3) and (3.4), we have
Since , it follows from (3.5) that
Now, we prove that .
Since, for any positive integer , it can be written as where , observe that
Since, for each , . Again since , we have
We observe that
Substituting (3.8), (3.9) in (3.7), we obtain
It follows from (3.2), (3.6) and (3.10) that
We also have
which gives that
For each , by Lemma 2.2(ii), is demiclosed at zero. This together with the fact that is bounded guarantees that every weak limit point of is a fixed point of (). That is . Since for we have for all , by the weak lower semi-continuity of the norm, we have
for all . However, since , we must have for all . Thus and then . Hence, by
This completes the proof. □
If we take in (1.14), then we obtain the following.
Corollary 3.2 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H. Let be a finite family of asymptotically -strictly pseudocontractive mappings of C into itself for some with Lipschitz constant , and for all such that and let . For and , assume that the control sequence is chosen such that , define as follows:
Then generated by (3.11) converges strongly to .
Corollary 3.3 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H. Let T be an asymptotically λ-strictly pseudocontractive mapping of C into itself such that , and let . For and , assume that the control sequence is chosen such that , define as follows:
Then generated by (3.12) converges strongly to .
Since asymptotically nonexpansive mappings are asymptotically 0-strict pseudocontractions, we have the following consequence.
Corollary 3.4 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H. Let T be an asymptotically nonexpansive mapping from C to itself such that , and let . For and , assume that the control sequence is chosen such that . Then generated by (3.12) converges strongly to .
Remark 3.5 From the main results, one can see that the corresponding results in Acedo and Xu , Inchan and Nammanee , Kim and Xu , Marino and Xu , Martinez-Yanez and Xu , Nakajo and Takahashi , Qin et al. , Yao and Chen  are all special cases of this paper.
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We are grateful to the referee for precise remarks which led to improvement of the paper. The second author is thankful to University Grants Commission of India for Project 41-1390/2012(SR).
The authors declare that they have no competing interests.
All authors contributed equally to this work. All authors read and approved the final manuscript.
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Cite this article
Dewangan, R., Thakur, B.S. & Postolache, M. A hybrid iteration for asymptotically strictly pseudocontractive mappings. J Inequal Appl 2014, 374 (2014). https://doi.org/10.1186/1029-242X-2014-374
- asymptotically strictly pseudocontractive mappings
- Mann iteration
- shrinking projection method
- strong convergence