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The equivalence between the convergence of the modified Mann and Ishikawa iterations for asymptotically pseudocontractive mappings obtained by dropping the bounded assumption
© Lv and Xue; licensee Springer 2014
Received: 12 August 2013
Accepted: 2 July 2014
Published: 18 August 2014
In this paper, we show the equivalence of convergence between the modified Mann and Ishikawa iterations with errors for an asymptotically pseudocontractive mapping under the condition of removing the bounded assumption. We also point out the problems of (Rhoades and Soltuz in J. Math. Anal. Appl. 283:681-688, 2003; Xue in Bull. Korean Math. Soc. 47(2):295-305, 2010; Xue in J. Math. Inequal. 4(3):345-354, 2010), extend and improve the results of (Zeng in Acta Math. Sin. 47(2):219-228, 2004).
MSC:47H10, 47H09, 46B20.
where denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by j.
Let D be a nonempty closed convex subset of E and be a mapping.
Definition 1.1 
- (1)T is called asymptotically nonexpansive if there exists a sequence with such that for all ,
- (2)T is called asymptotically pseudocontractive if there exists a sequence with such that for all , there exists ,
Remark 1.2 
T is an asymptotically pseudocontractive map;
- (ii)there exists with such that(1.1)
Obviously, the asymptotically pseudocontractive and asymptotically nonexpansive mappings with the constant sequence are the usual definition of strongly pseudocontractive and nonexpansive mappings, respectively. An asymptotically nonexpansive mapping is asymptotically pseudocontractive. The converse is not true in general; see . And it is clear that an asymptotically nonexpansive mapping is also uniformly L-Lipschitz for some , where .
Let us recall some iterations in the following.
where , are any bounded sequences of D. , , , are four real sequences in satisfying and for any .
Recently, many authors [2, 6–8] have proved the iterative approximation problem of fixed point for uniformly L-Lipschitz asymptotically pseudocontractive mappings in Banach spaces. The results are as follows.
Theorem 1.5 (, Theorem 2.1)
converges strongly to q.
Theorem 1.6 (, Theorem 2.2)
the Mann iteration with errors converges strongly to the fixed point of ;
the Ishikawa iteration with errors converges strongly to the fixed point of .
Remark 1.7 There exists a gap in the proof process of Theorem 2.1 of . It is in lines 3-4 of P300, ‘’, where is an infinite subsequence of the sequence . Meanwhile, there exists a similar problem in Theorem 2.2 of  (for more details, see 11th of P303). For this, we provide an example. Let , , , then , but there does not exist any subsequence of the sequence such that . Hence we cannot obtain that , , . So Theorems 2.1, 2.2 of  do not hold.
Theorem 1.8 (, Theorem 8)
the modified Mann iteration (1.5) converges to ;
the modified Ishikawa iteration (1.4) converges to .
does not hold. The reason is . By remark (1.2), the result of does not hold.
In 2004, Zeng  gave another result as follows.
Theorem 1.9 (, Theorem 3.1)
then the modified Ishikawa iteration with errors converges strongly to .
But this result is not perfect because of the assumption of bounded range.
The aim of this paper is to revise the results of the papers [2, 6, 7] and remove the assumption T with bounded range . We obtain that the modified Ishikawa iteration with errors converges strongly to the fixed point of T and the modified Mann and Ishikawa iterations with errors are equivalent. For these, we need the following lemmas.
Lemma 1.10 
for all .
Lemma 1.11 
If , , then exists.
where . If exists, then as .
From (iv) and (1.11), we have which is a contradiction to condition (ii) and so , i.e., . □
2 Main results
the modified Mann iteration with errors converges strongly to ;
the modified Ishikawa iteration with errors converges strongly to .
Proof If the modified Ishikawa iteration with errors sequence defined by (1.2) converges strongly to q, then setting , , we obtain the convergence of the modified Mann iteration with errors sequence defined by (1.3). Conversely, we only prove that .
Since , , then , are bounded. Set .
First we prove that the sequence is bounded.
where = + + + + + , = + + + and satisfy , . By Lemma 1.11, exists. Hence the sequence is bounded. Set .
Since , , then , . By (iii) and (iv), we have . Using Lemma 1.12, we obtain that . □
where , then the modified Ishikawa iteration with errors converges strongly to .
Proof In the proof course of Theorem 2.1, setting , for , we obtain Theorem 2.2. □
It is worth mentioning that the result extends Theorem 3.1 in  by dropping the bounded assumption. See the following example.
Then the mapping T satisfies Theorem 2.2. But the range of T is not bounded.
where , then the modified Mann iteration converges strongly to .
Proof In Theorem 2.2, setting , we obtain Corollary 2.4. □
The control conditions of the parameters in Corollary 2.4 are different from those of Theorem 2.1 of . See the following example.
Then as , and , , , but as does not hold.
We remove the hypothesis T with bounded range and obtain the same result by the different method from .
We extend formula (2.1) of  to (2.13) in this paper.
We also obtain the equivalence between the convergence of the modified Mann iteration with errors and the modified Ishikawa iteration with errors for an asymptotically pseudocontractive mapping.
The authors acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 11372196) and the Shijiazhuang Tiedao University Foundation (Grant No. 20133026).
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