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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
Journal of Inequalities and Applications volume 2014, Article number: 293 (2014)
Abstract
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.
1 Introduction
Let E be a real Banach space and be its dual space. The normalized duality mapping is defined by
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]
-
(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 [2]
It is very well known that the following conditions are equivalent:
-
(i)
T is an asymptotically pseudocontractive map;
-
(ii)
there exists with such that
(1.1)
Definition 1.3 A mapping T is called uniformly L-Lipschitzs if there exists such that for any ,
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 [3]. 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.
Definition 1.4 For arbitrary given , the modified Ishikawa iteration with errors is defined by
where , are any bounded sequences of D. , , , are four real sequences in satisfying and for any .
If for all , then (1.2) reduces to the modified Mann iteration with errors as follows:
If for any , then for , (1.2) and (1.3) reduce to the modified Ishikawa and Mann iterations as follows, respectively (see [4] and [5]):
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 ([6], Theorem 2.1)
Let E be a real Banach space, D be a nonempty closed convex subset of E and be a uniformly -Lipschitzian asymptotically Φ-pseudocontractive mapping with the sequence , . Let . Let and satisfy , and with for all . Then the Mann iterative process with errors defined by
converges strongly to q.
Theorem 1.6 ([6], Theorem 2.2)
Let E be a real Banach space, D be a nonempty closed convex subset of E and () be two uniformly L-Lipschitzian asymptotically Φ-pseudocontractive mappings with the sequences such that and . Let , be two sequences in satisfying the conditions: (i) ; (ii) . Then the following two assertions are equivalent:
-
(i)
the Mann iteration with errors converges strongly to the fixed point of ;
-
(ii)
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 [6]. 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 [6] (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 [6] do not hold.
Theorem 1.8 ([2], Theorem 8)
Let X be a real Banach space, B be a nonempty closed convex subset of X and , be defined by (1.5) and (1.4) with , satisfying the following conditions: , , . Let T be an asymptotically pseudocontractive and uniformly L-Lipschitzian with self-map of B. Let be the fixed point of T. If , then the following two assertions are equivalent:
-
(i)
the modified Mann iteration (1.5) converges to ;
-
(ii)
the modified Ishikawa iteration (1.4) converges to .
But there exists an error in the proof course for the above theorem, i.e., P 684 the following formula
does not hold. The reason is . By remark (1.2), the result of [2]does not hold.
In 2004, Zeng [8] gave another result as follows.
Theorem 1.9 ([8], Theorem 3.1)
Let E be a real Banach space, D be a nonempty closed convex subset of E and be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with the sequence , . Suppose that is defined by (1.2), where , , , are four real number sequences in satisfying the following conditions:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, ;
-
(iv)
, .
Suppose that the range of T is bounded and . If there exists a strictly increasing continuous function with such that
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 [8]. 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 [9]
Let E be a real Banach space and let be a normalized duality mapping. Then
for all .
Lemma 1.11 [10]
Let , , be three nonnegative real sequences satisfying the inequality
If , , then exists.
Lemma 1.12 Let , , , , and be six nonnegative real sequences satisfying the following conditions:
-
(i)
;
-
(ii)
;
-
(iii)
, ;
-
(iv)
, .
Let be a strictly increasing and continuous function with such that
where . If exists, then as .
Proof Since exists, we define and . We declare that . If it is not this case, then , there exists a natural number such that for . Since Φ is strictly increasing, then . From condition (iii), we obtain that there exists such that , for . By (1.8), we have
which implies that
It leads to
From (iv) and (1.11), we have which is a contradiction to condition (ii) and so , i.e., . □
2 Main results
Theorem 2.1 Let D be a nonempty closed convex subset of the real Banach space E. Suppose that is a uniformly L-Lipschitz asymptotically pseudocontractive mapping with the real number sequence , . Let and be defined by (1.2) and (1.3), respectively, where , , and are four real number sequences in satisfying the following conditions:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, ;
-
(iv)
, .
Suppose . If there exists a strictly increasing continuous function with such that
where , then the following two assertions are equivalent:
-
(1)
the modified Mann iteration with errors converges strongly to ;
-
(2)
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.
From (1.2) and (1.3), we have
and
Using (2.2) and (2.3), we have
Since T satisfies (2.1), so T is an asymptotically pseudocontractive map. Applying (1.1), we get
which implies that
From (1.2) and (1.3), we obtain the following inequalities:
Taking (2.6) into (2.7), we obtain that
From (1.3), we get
Substituting (2.6), (2.8) and (2.9) into (2.5), we have
where = + + + + + , = + + + and satisfy , . By Lemma 1.11, exists. Hence the sequence is bounded. Set .
It follows from (1.2), (1.3), (2.1) and Lemma 1.10 that we have
where = + + . Since , then there exists such that for . So (2.11) becomes
Since , , then , . By (iii) and (iv), we have . Using Lemma 1.12, we obtain that . □
Theorem 2.2 Let D be a nonempty closed convex subset of the real Banach space E. Suppose that is a uniformly L-Lipschitz asymptotically pseudocontractive mapping with the real number sequence , . Suppose that is defined by (1.2), where , , , are four real number sequences in satisfying the following conditions:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, ;
-
(iv)
, .
Suppose . If there exists a strictly increasing continuous function with such 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 [8] by dropping the bounded assumption. See the following example.
Example 2.3 Let be a real space with the usual norm. Define by , , , , . Then Φ is a strictly increasing continuous function with and T has a fixed point . For any , we obtain that
Then the mapping T satisfies Theorem 2.2. But the range of T is not bounded.
Corollary 2.4 Let D be a nonempty closed convex subset of the real Banach space E. Suppose that is a uniformly L-Lipschitz asymptotically pseudocontractive mapping with the real number sequence , . Let be defined by (1.3), where , are two real number sequences in satisfying the following conditions:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
.
Suppose . If there exists a strictly increasing continuous function with such that
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 [6]. See the following example.
Example 2.5 Set , , , . Then as and , but , and . On the other hand, let
Then as , and , , , but as does not hold.
Remark 2.6 Our theorems extend and improve the corresponding results of [2, 6–8] in the following sense:
- (1)
-
(2)
We remove the hypothesis T with bounded range and obtain the same result by the different method from [8].
-
(3)
We extend formula (2.1) of [8] to (2.13) in this paper.
-
(4)
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.
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Acknowledgements
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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Lv, G., Xue, Z. The equivalence between the convergence of the modified Mann and Ishikawa iterations for asymptotically pseudocontractive mappings obtained by dropping the bounded assumption. J Inequal Appl 2014, 293 (2014). https://doi.org/10.1186/1029-242X-2014-293
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DOI: https://doi.org/10.1186/1029-242X-2014-293