Abstract
Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and be a generalized Lipschitz Φhemicontractive mapping with . Let , , , be four real sequences in and satisfy the conditions (i) as and ; (ii) . For some , let , be any bounded sequences in D, and be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φquasiaccretive mapping.
MSC: 47H10.
Keywords:
generalized Lipschitz mapping; Φhemicontractive mapping; Ishikawa iterative sequence with errors; uniformly smooth real Banach space1 Introduction and preliminary
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. It is well known that
(i) If E is a smooth Banach space, then the mapping J is singlevalued;
(iii) If E is a uniformly smooth Banach space, then the mapping J is uniformly continuous on any bounded subset of E. We denote the singlevalued normalized duality mapping by j.
Definition 1.1 ([1])
Let D be a nonempty closed convex subset of E, be a mapping.
(1) T is called strongly pseudocontractive if there is a constant such that for all ,
(2) T is called ϕstrongly pseudocontractive if for all , there exist and a strictly increasing continuous function with such that
(3) T is called Φpseudocontractive if for all , there exist and a strictly increasing continuous function with such that
It is obvious that Φpseudocontractive mappings not only include ϕstrongly pseudocontractive mappings, but also strongly pseudocontractive mappings.
Definition 1.2 ([1])
(1) T is called ϕstronglyhemipseudocontractive if for all , , there exist and a strictly increasing continuous function with such that
(2) T is called Φhemipseudocontractive if for all , , there exist and the strictly increasing continuous function with such that
Closely related to the class of pseudocontractivetype mappings are those of accretive type.
Definition 1.3 ([1])
Let . The mapping is called strongly quasiaccretive if for all , , there exist and a constant such that ; T is called ϕstrongly quasiaccretive if for all , , there exist and a strictly increasing continuous function with such that ; T is called Φquasiaccretive if for all , , there exist and a strictly increasing continuous function with such that .
Definition 1.4 ([2])
For arbitrary given , the Ishikawa iterative process with errors is defined by
where , are any bounded sequences in D; , , , are four real sequences in and satisfy , , for all . If , then the sequence defined by
is called the Mann iterative process with errors.
A mapping is called generalized Lipschitz if there exists a constant such that , .
The aim of this paper is to prove the convergent results of the above Ishikawa and Mann iterations with errors for generalized Lipschitz Φhemicontractive mappings in uniformly smooth real Banach spaces. For this, we need the following lemmas.
Lemma 1.6 ([5])
LetEbe a uniformly smooth real Banach space, and letbe a normalized duality mapping. Then
Lemma 1.7 ([6])
Letbe a nonnegative sequence which satisfies the following inequality:
2 Main results
Theorem 2.1LetEbe an arbitrary uniformly smooth real Banach space, Dbe a nonempty closed convex subset ofE, andbe a generalized Lipschitz Φhemicontractive mapping with. Let, , , be four real sequences inand satisfy the conditions (i) asand; (ii) . For some, let, be any bounded sequences inD, andbe an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the unique fixed pointqofT.
Proof Since is a generalized Lipschitz Φhemicontractive mapping, there exists a strictly increasing continuous function with such that
i.e.,
and
Step 1. There exists and such that (range of Φ). Indeed, if as , then ; if with , then for , there exists a sequence in D such that as with . Furthermore, we obtain that is bounded. Hence, there exists a natural number such that for , then we redefine and .
Step 2. For any , is bounded. Set , then from Definition 1.2(2), we obtain that . Denote , . Since T is generalized Lipschitz, so T is bounded. We may define . Next, we want to prove that . If , then . Now, assume that it holds for some n, i.e., . We prove that . Suppose it is not the case, then . Since J is uniformly continuous on a bounded subset of E, then for , there exists such that when , . Now, denote
Owing to as , without loss of generality, assume that for any . Since , denote . So, we have
and
Therefore,
Using Lemma 1.6 and the above formulas, we obtain
and
Substitute (2.11) into (2.10)
this is a contradiction. Thus, , i.e., is a bounded sequence. So, , , are all bounded sequences.
Step 3. We want to prove as . Set .
By (2.10), (2.11), we have
and
Taking (2.14) into (2.13),
Set , then . If it is not the case, we assume that . Let , then , i.e., . Thus, from (2.15) it follows that
This implies that
Applying Lemma 1.7, we get that as . This is a contradiction and so . Therefore, there exists an infinite subsequence such that as . Since , then as . In view of the strictly increasing continuity of Φ, we have as . Hence, as . Next, we want to prove as . Let , there exists such that , , , , for any . First, we want to prove . Suppose it is not the case, then . Using (1.1), we may get the following estimates:
Since Φ is strictly increasing, then (2.19) leads to . From (2.15), we have
which is a contradiction. Hence, . Suppose that holds. Repeating the above course, we can easily prove that holds. Therefore, for any m, we obtain that , which means as . This completes the proof. □
Theorem 2.2LetEbe an arbitrary uniformly smooth real Banach space, andbe a generalized Lipschitz Φquasiaccretive mapping with. Let, , , be four real sequences inand satisfy the conditions (i) asand; (ii) . For some, let, be any bounded sequences inE, andbe an Ishikawa iterative sequence with errors defined by
whereis defined byfor any. Thenconverges strongly to the unique solution of the equation (or the unique fixed point ofS).
Proof Since T is a generalized Lipschitz and Φquasiaccretive mapping, it follows that
i.e.,
i.e.,
for all , . The rest of the proof is the same as that of Theorem 2.1. □
Corollary 2.3LetEbe an arbitrary uniformly smooth real Banach space, Dbe a nonempty closed convex subset ofE, andbe a generalized Lipschitz Φhemicontractive mapping with. Let, be two real sequences inand satisfy the conditions (i) asand; (ii) . For some, letbe any bounded sequence inD, andbe the Mann iterative sequence with errors defined by (1.2). Then (1.2) converges strongly to the unique fixed pointqofT.
Corollary 2.4LetEbe an arbitrary uniformly smooth real Banach space, andbe a generalized Lipschitz Φquasiaccretive mapping with. Let, be two real sequences inand satisfy the conditions (i) asand; (ii) . For some, letbe any bounded sequence inE, andbe the Mann iterative sequence with errors defined by
whereis defined byfor any. Thenconverges strongly to the unique solution of the equation (or the unique fixed point ofS).
Remark 2.5 It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φquasiaccretive operators. This result provided significant improvements of some recent important results. Their result is as follows.
Theorem CC ([[1], Theorem 3.1])
LetEbe a uniformly smooth real Banach space andbe a mapping with. SupposeAis a generalized Lipschitz Φquasiaccretive mapping. Let, andbe real sequences insatisfying the following conditions: (i) ; (ii) ; (iii) ; (iv) . Letbe generated iteratively from arbitraryby
whereis defined by, andis an arbitrary bounded sequence inE. Then, there existssuch that if, , the sequenceconverges strongly to the unique solution of the equation.
However, there exists a gap in the proof process of above Theorem CC. Here, () does not hold in line 14 of Claim 2 on page 248, i.e., is a wrong case. For instance, set the iteration parameters: , where , , ; . Then , but . Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in writing this paper. All authors read and approved the final manuscript.
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