Abstract
In this paper, we prove some strong and weak convergence theorems for continuous pseudocontractive mapping and a weak convergence theorem for nonexpansive mapping in real uniformly convex Banach spaces. As an application of the strong convergence theorem, we give an interesting example.
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1 Introduction and preliminaries
Throughout this paper, we assume that E is a real Banach space, is the dual space of E and is the normalized duality mapping defined by
for all , where denotes duality pairing between E and . A single-valued normalized duality mapping is denoted by j.
Let C be a nonempty closed convex subset of a real Banach space E. A mapping is said to be pseudocontractive [1] if, for any , there exists such that
It is well known [2] that (1.1) is equivalent to the following:
for all and .
A mapping is said to be nonexpansive if
for all .
Remark 1.1 It is easy to see that, if T is nonexpansive, then T is continuous pseudocontractive, but the converse is not true in general.
Example 1.1 Let with the usual norm . Then , for all and and the normalized duality mapping is as follows:
for all . Let and define a mapping by for all . It is easy to prove that T is continuous pseudocontractive, but not nonexpansive.
In 1974, Deimling [3] proved the following fixed point theorem.
Theorem 1.1 Let E be a real Banach space, C be a nonempty closed convex subset of E, and be a continuous strongly pseudocontractive mapping. Then T has a unique fixed point in C.
Let E be a real Banach space, C be a nonempty closed convex subset of E and be a continuous pseudocontractive mapping. For all and , define the mapping by
for all . It is easy to prove that is a continuous strongly pseudocontractive mapping. By Theorem 1.1, there exists a unique fixed point of such that
for all .
Let be a continuous pseudocontractive mapping and define an implicit iteration process by
for each , where is a sequence in with some conditions.
In 2008, Zhou [4] studied the implicit iteration process (1.3) and proved a weak convergence theorem for the strict pseudocontraction in a real reflexive Banach space which satisfies Opial’s condition.
The purpose of this paper is to discuss the implicit iteration process (1.3) and to prove some strong and weak convergence theorems for a continuous pseudocontractive mapping and a weak convergence theorem for a nonexpansive mapping in real uniformly convex Banach spaces. As an application of the strong convergence theorem, we give an interesting example.
In order to prove the main results, we need the following:
A Banach space E is said to satisfy Opial’s condition [5] if, for any sequence of E, weakly as implies that
for all with . A Banach space E is said to have the Kadec-Klee property [6] if, for every sequence in E, weakly and , it follows that strongly.
Lemma 1.1 ([7])
Let E be a uniformly convex Banach space with the modulus of uniform convexity . Then is continuous, increasing, , for and, further,
whenever and .
Lemma 1.2 ([8])
Let X be a uniformly convex Banach space and C be a convex subset of X. Then there exists a strictly increasing continuous convex function with such that, for each with Lipschitz constant L,
for all and .
Lemma 1.3 ([8])
Let X be a uniformly convex Banach space such that its dual space has the Kadec-Klee property. Suppose that is a bounded sequence and (where denotes the set of all weak subsequential limits of a bounded sequence in X) such that
exists for all . Then .
Lemma 1.4 ([4])
Let E be a real uniformly convex Banach space, C be a nonempty closed convex subset of E and be a continuous pseudocontractive mapping. Then is semi-closed at zero, i.e., for each sequence in C, if converges weakly to and converges strongly to 0, then .
2 Convergence for continuous pseudocontractive mappings
Lemma 2.1 Let E be a real uniformly convex Banach space, C be a nonempty closed convex subset of E and be a continuous pseudocontractive mapping with . Let be defined by (1.3), where and . Then
-
(1)
for all and all ;
-
(2)
exists for all ;
-
(3)
exists;
-
(4)
.
Proof For all and , since T is pseudocontractive, it follows from (1.3) that
which implies that and . Therefore, and exist. Thus (1), (2), and (3) are proved.
Now, we prove (4). By using (1.2), (1.3), and Lemma 1.1, it follows that, for all ,
This implies that
If , then we have as by the properties of . It follows from and that as and so
as .
If , then, since T is continuous, it follows that
This completes the proof. □
Theorem 2.1 Under the assumptions of Lemma 2.1, converges strongly to a fixed point of T if and only if .
Proof The necessity is obvious. So, we will prove the sufficiency. Assume that
By Lemma 2.1, limit exists and so .
Now, we show that is a Cauchy sequence in C. In fact, it follows from Lemma 2.1 that for all positive integers m, n with and . So,
Taking the infimum over all , we have
It follows from that is a Cauchy sequence. C is a closed subset of E and so converges strongly to some . Further, by the continuity of T, it is easy to prove that is closed and it follows from that . This completes the proof. □
Corollary 2.1 Under the assumptions of Lemma 2.1, converges strongly to a fixed point p of T if and only if there exists a subsequence of such that converges strongly to p.
Proof Since
it follows from Theorem 2.1 that Corollary 2.1 holds. This completes the proof. □
Theorem 2.2 Under the assumptions of Lemma 2.1, if there exists a nondecreasing function with and for all such that
for all , then converges strongly to a fixed point of T.
Proof Since by Lemma 2.1 and so . Further, by using Lemma 2.1 exists, and we assume .
If , there exists a positive integer N such that for all . Thus we have
which is a contradiction. Therefore, . It follows from Theorem 2.1 that Theorem 2.2 holds. This completes the proof. □
Definition 2.1 ([9])
Let E be a real normed linear space, C be a nonempty subset of E, and be a mapping. The pair is said to satisfy the condition (A) if, for any bounded closed subset G of C, is a closed subset of E.
Now, we prove strong convergence and weak convergence theorems for a continuous pseudocontractive mapping in real uniformly convex Banach spaces.
Theorem 2.3 Under the assumptions of Lemma 2.1, if the pair satisfies the condition (A), then converges strongly to a fixed point of T.
Proof The sequence is bounded in C by Lemma 2.1. Letting , where denotes the closure of A, G is a bounded closed subset of C and so is a closed subset of E since the pair satisfies the condition (A). It follows from and as by Lemma 2.1(3) that the zero vector and so there exists such that . This shows that q is a fixed point of T and so there exists a positive integer such that or there exists a subsequence of such that as .
If , then it follows from Lemma 2.1(1) that for all and so as .
If , then, since exists by Lemma 2.1(2), as . This completes the proof. □
As an application of Theorem 2.1, we give the following.
Example 2.1 Let with the usual norm . Then , for all and and for all . Let . Define a mapping by
Then T is continuous pseudocontractive with and the pair satisfies the condition (A). In fact, for all and , we have and so
For all and , we obtain and so
Thus, for all , taking , it follows from (2.1) and (2.2) that
This shows that T is continuous pseudocontractive.
Now, we prove that the pair satisfies the condition (A). For any bounded closed subset G of C, we denote . Then M is closed. Indeed, for any with , there exists such that . We consider the following cases.
Case 1. There exists a positive integer such that for all .
Case 2. There exists a subsequence of such that for all .
If Case 1 holds, then for all and so as . Since G is closed, it follows that and so .
If Case 2 holds, then for all and so . If , we have . Otherwise, we have and so .
By Theorem 1.1, it is easy to prove that, for any , there exists a unique such that
for all , where for all and
Thus it follows from Theorem 2.3 that the sequence converges to .
Theorem 2.4 Under the assumptions of Lemma 2.1, if E satisfies Opial’s condition, then converges weakly to a fixed point of T.
Proof Since is bounded by Lemma 2.1 and E is reflexive, there exists a subsequence of which converges weakly to a point . By Lemma 2.1, we have . It follows from Lemma 1.4 that .
Now, we prove that converges weakly to p. Suppose that there exists a subsequence of such that converges weakly to a point . Then . In fact, if , then it follows from Opial’s condition that
which is a contradiction. So . Therefore, converges weakly to a fixed point of T. This completes the proof. □
Remark 2.1 By Remark 1.1, clearly, Theorems 2.1, 2.2, 2.3 and 2.4 still hold for nonexpansive mappings.
3 Weak convergence for nonexpansive mappings
In this section, we prove a weak convergence theorem for a nonexpansive mapping in real uniformly convex Banach spaces.
Lemma 3.1 Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let be a nonexpansive mapping with and be the sequence defined by (1.3), where and . Then, for all , the limit exists for all .
Proof Letting for all , and exists by Remark 1.1 and Lemma 2.1. Thus it remains to prove Lemma 3.1 for any . For all and , we define a mapping by
for all . Then we have
for all . It follows from that is contractive and so it has a unique fixed point in C, which is denoted by . Define a mapping by
where I is a identity mapping. It follows from (3.1) that
for all and so
Using (3.1) and (1.3), we obtain
and
for all . It follows from (3.3), (3.4) and that and . For each , let
and
By (3.2), we have
for all . This shows that is nonexpansive, and for all . By Lemma 1.2, we obtain
It is easy to prove that
For fixed n and letting , we have
Again, letting , we obtain
This shows that
exists for all . This completes the proof. □
Theorem 3.1 Under the assumptions of Lemma 3.1, if the dual space of E has the Kadec-Klee property, then converges weakly to a fixed point of T.
Proof Using the same method as in the proof of Theorem 2.4, we can prove that there exists a subsequence of , which converges weakly to a point .
Now, we prove that converges weakly to p. Suppose that there exists a subsequence of such that converges weakly to a point . Then . In fact, it follows from Lemma 3.1 that the limit
exists for all . Again, since , we have by Lemma 1.3. This shows that converges weakly to p. This completes the proof. □
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Acknowledgements
The project was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).
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Guo, W., Choi, M.S. & Cho, Y.J. Convergence theorems for continuous pseudocontractive mappings in Banach spaces. J Inequal Appl 2014, 384 (2014). https://doi.org/10.1186/1029-242X-2014-384
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DOI: https://doi.org/10.1186/1029-242X-2014-384