A hybrid iteration for asymptotically strictly pseudocontractive mappings
© Dewangan et al.; licensee Springer. 2014
Received: 2 May 2014
Accepted: 2 September 2014
Published: 29 September 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.
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.
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 .
We now give an example to show that a λ-strictly asymptotically pseudocontractive mapping is not necessarily a λ-strictly pseudocontractive mapping.
for all , . Since , it follows that T is an asymptotically strictly pseudocontractive mapping.
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.
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 .
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.
where and are given by formulas (1.9) and (1.10), respectively.
where , .
and is given by the same formula (1.9).
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.
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.
⇀ for weak convergence and → for strong convergence.
denotes the weak ω-limit set of .
the set of fixed points of T.
Lemma 2.1 ()
Lemma 2.2 ()
- (i)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 ()
is convex (and closed).
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.
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.
Hence is bounded.
Next, we show that exists.
Hence is a Cauchy sequence, and so convergent.
Now, we prove that .
This completes the proof. □
If we take in (1.14), then we obtain the following.
Then generated by (3.11) converges strongly to .
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.
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).
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