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A further remark to paper ‘Convergence theorems for the common solution for a finite family of ϕ-strongly accretive operator equations’
Journal of Inequalities and Applications volume 2013, Article number: 35 (2013)
Abstract
In this note, we point out several gaps in Gurudwan and Sharma (Appl. Math. Comput. 217(15):6748-6754, 2011) and Yang (Appl. Math. Comput. 218(21):10367-10369, 2012) and give the main results under weaker conditions.
MSC:47H10, 47H09, 46B20.
1 Introduction
Recently, Gurudwan, Sharma [1] and Yang [2] studied the strong convergence of the sequence, respectively, which was defined by
for approximation of a common solution of a finite family of uniformly continuous Φ-strongly accretive operator equations. Their results are as follows.
Theorem GS [[1], Theorem 3.1]
Let E be an arbitrary real Banach space and let be uniformly continuous ϕ-strongly accretive operators and each range of either or be bounded. Let, for , be sequences in E and , , be real sequences in satisfying
-
(i)
,
-
(ii)
,
-
(iii)
,
-
(iv)
, , .
For any given , define by , , . Then the multi-step iterative sequence with errors defined by the above converges strongly to the unique solution of the operator equations .
On the basis of the above result, Yang [2] proved the following convergence theorem.
Thoerem Yang [[2], Theorem 2]
Let E be an arbitrary real Banach space and let be uniformly continuous ϕ-strongly accretive operators and each range of either or is bounded. Let for , be bounded sequences in E and , , be real sequences in satisfying
-
(i)
,
-
(ii)
,
-
(iii)
, , .
For any given , define by , , . Then the multi-step iterative sequence with errors defined by the above converges strongly to the unique solution of the operator equations .
However, after careful reading of their works, we discovered that there exist some problems in references [1] and [2] as follows.
Problem 1 In the proof course of Theorem 3.1 of Gurudwan and Sharma [1], which happens in line 11 of page 6751. Here, it is defective that they obtained , that is, , but we cannot deduce . The reason is that it is possible does not belong to (range of ϕ). A counterexample is as follows. Let us define by ; then it can be easily seen that ϕ is increasing with , but and makes no sense (see [3]).
Problem 2 In the paper of Yang [2], he referred to the mistakes of ‘ for to deduce ()’ in [1] and cited an example, i.e.,
Now, we want to clarify the fact. Let be a real sequence, be some infinite subsequence of and be neither odd nor even sequence, then the conclusions are as follows:
(C-i) , ∃ nonnegative integer such that for , .
(C-ii) and for .
Indeed, the above example (∗∗) does not satisfy the conclusion (C-i), it just illustrates the result (C-ii). Therefore, the note given by Yang [2] confused the conclusions (C-i) and (C-ii).
The aim of this paper is to generalize the results of papers [1] and [2]. For this, we need the following knowledge.
2 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. The single-valued normalized duality mapping is denoted by j.
An operator is said to be strongly accretive if there exists a constant , and for , such that
without loss of generality, we assume that . The operator T is called ϕ-strongly accretive if for any , there exist and a strictly increasing continuous function with such that
It is obvious that a strongly accretive operator must be the ϕ-strongly accretive in the special case in which , but the converse is not true in general. That is, the class of strongly accretive operators is a proper subclass of the class of ϕ-strongly accretive operators.
In order to obtain the main conclusion of this paper, we need the following lemmas.
Lemma 2.1 [1]
Suppose that E is an arbitrary Banach space and is a continuous ϕ-strongly accretive operator. Then the equation has a unique solution for any .
Lemma 2.2 [4]
Let E be a real Banach space and let be a normalized duality mapping. Then
for all and .
Lemma 2.3 [5]
Let , and be three nonnegative real sequences and be a strictly increasing and continuous function with satisfying the following inequality:
where with , . Then as .
3 Main results
Theorem 3.1 Let E be an arbitrary real Banach space and be N uniformly continuous ϕ-strongly accretive operators. For , let be bounded sequences in E and , , be real sequences in satisfying
-
(i)
, ;
-
(ii)
;
-
(iii)
, ;
-
(iv)
.
For any given , define with by , , , where . Then, for some , the multi-step iterative sequence with errors defined by
converges strongly to the unique solution of the operator equations .
Proof Since is ϕ-strongly accretive operator, we obtain that each equation has the unique solution by Lemma 2.1, denote , i.e., is the unique fixed point of by . Since , then is a single set, let q. Meanwhile, there exists a strictly increasing continuous function with such that
for , , that is,
Choose some and such that , where
is the range of Φ. Indeed, if as , then ; if with , then for , there exists a sequence in E such that as with . Since is uniformly continuous, so is . Furthermore, we obtain that as , then is the bounded sequence for . Hence, there exists the common natural number such that for and , then we redefine and . Thus, . It is to ensure that is defined well.
Step I. We show that is a bounded sequence.
Set , then from the above formula (@), we obtain that . Denote
Since is uniformly continuous, then is bounded. We let
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 is uniformly continuous for , then for , there exists common such that when . Denote
Since as for . Without loss of generality, we let for any and . Since , let . Now, estimate for . From the multi-step iteration, we have
then . Similarly, we have
then . ⋯⋯ , we have
then . Therefore, we get
And we also have
and
By the uniform continuity of , we have
Using Lemma 2.2 and the above formulas, we have
which is a contradiction. So, , i.e., is a bounded sequence, from which it follows that are all bounded sequences as well.
Step II. We want to prove as .
Since as for and , are bounded. From (3.5) and (3.6), we obtain
By (3.7), we have
where
By Lemma 2.3, we obtain . This completes the proof. □
Remark 3.2 Theorem 3.1 generalizes Theorem 3.1 of [1] and Theorem 2 of [2] in the following cases:
-
(a)
It is not necessary for each range of or to be bounded in [1] and [2].
-
(b)
The condition of is weakened to from ().
-
(c)
The proof method of our theorem differs from that of [1] and [2].
Theorem 3.3 Let E, , , , () be as in Theorem 3.1 and let be N uniformly continuous ϕ-strongly pseudocontractive mappings. Then, for some , the multi-step iterative sequence with errors defined by
converges strongly to the unique common fixed point of .
Proof See [1]. □
References
Gurudwan N, Sharma BK: Convergence theorem for the common solution for a finite family of ϕ -strongly accretive operator equations. Appl. Math. Comput. 2011, 217(15):6748–6754. 10.1016/j.amc.2011.01.093
Yang L: A note on a paper ‘Convergence theorem for the common solution for a finite family of ϕ -strongly accretive operator equations’. Appl. Math. Comput. 2012, 218(21):10367–10369. 10.1016/j.amc.2012.04.037
Rafiq A: On iterations for families of asymptotically pseudocontractive mappings. Appl. Math. Lett. 2011, 24(1):33–38. 10.1016/j.aml.2010.08.005
Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.
Moore C, Nnoli BVC: Iterative solution of nonlinear equations involving set-valued uniformly accretive operators. Comput. Math. Appl. 2001, 42(1–2):131–140. 10.1016/S0898-1221(01)00138-9
Acknowledgements
The authors are very grateful to Professor Yeol-Je Cho for good suggestions which helped to improve the manuscript. This work is supported by the Hebei Natural Science Foundation No. A2011210033.
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Xue, Z., Zhou, H. A further remark to paper ‘Convergence theorems for the common solution for a finite family of ϕ-strongly accretive operator equations’. J Inequal Appl 2013, 35 (2013). https://doi.org/10.1186/1029-242X-2013-35
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DOI: https://doi.org/10.1186/1029-242X-2013-35