Strong convergence theorems for asymptotically nonexpansive nonself-mappings with applications
- Weiping Guo^{1},
- Afrah AN Abdou^{2}Email author,
- Liaqat A Khan^{2} and
- Yeol Je Cho^{2, 3}Email author
https://doi.org/10.1186/s13663-015-0463-6
© Guo et al. 2015
Received: 2 May 2015
Accepted: 6 November 2015
Published: 18 November 2015
Abstract
In this paper, first, we introduce the condition (BP) which is weaker than the completely continuous mapping in Banach spaces. Second, we consider a simple iteration and prove some strong convergence theorems of the proposed iteration for an asymptotically nonexpansive nonself-mapping with the condition (BP). Finally, we give two examples to illustrate the main result in this paper. Our results improve and extend the corresponding results given by some authors.
Keywords
1 Introduction
In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive self-mappings.
Theorem GK
If C is a nonempty closed convex subset of a real uniformly convex Banach space E and \(T:C\rightarrow C\) is an asymptotically nonexpansive self-mapping, then T has a fixed point in C.
On the other hand, in 1991, Schu [2] introduced the modified Mann process to approximate fixed points of an asymptotically nonexpansive self-mapping defined on a nonempty closed convex and bounded subset C of a Hilbert space H as follows:
Theorem JS
In Theorem JS, the mapping T remains a self-mapping of a nonempty closed convex subset C of a Hilbert space H. If, however, the domain \(D(T)\) of T is a proper subset of H and \(T: D(T )\rightarrow H\) is a mapping, the modified iteration \(\{x_{n}\}\) defined by (JS) may fail to be well defined.
To overcome this situation, in 2003, Chidume et al. [3] introduced the concept of asymptotically nonexpansive nonself-mappings.
Let E be a real Banach space. A subset C of E is called a retract of E if there exists a continuous mapping \(P:E\rightarrow K\) such that \(Px=x\) for all \(x\in C\). Every closed convex subset of a uniformly convex Banach space is a retract. A mapping \(P:E\rightarrow E\) is called a retraction if \(P^{2}=P\). It follows that, if a mapping P is a retraction, then \(Py=y\) for all y in the range of P (see [3]).
Definition 1.1
Remark 1.1
Since the results of Chidume et al., some authors proved weak and strong convergence theorems for asymptotically nonexpansive nonself-mappings in Banach spaces (see [4–7]).
In this paper, first, we introduce the condition (BP) which is weaker that the completely continuous mapping. Second, we introduce a new iteration (1.2) and prove some strong convergence theorems of the proposed iteration for an asymptotically nonexpansive nonself-mapping \(T:C\rightarrow E\) with the condition (BP). Finally, we give two examples to illustrate the main result in this paper. Our results improve and extend the corresponding results given by some authors.
2 Some lemmas
For our main results, we need the following lemmas.
Lemma 2.1
([8])
Letting \(S_{1}=S_{2}=I\), where I denotes the identity mapping, and \(T_{1}=T_{2}=T\) in Lemma 3.1 of [4], we have the following.
Lemma 2.2
- (1)
\(\|x_{n+1}-p\|\leq h_{n}^{2}\|x_{n}-p\|\) for all \(p\in F(T)\);
- (2)
\(\lim_{n\rightarrow\infty}\|x_{n}-p\|\) exists for all \(p\in F(T)\).
Lemma 2.3
Proof
3 The condition (BP)
Let E be a Banach space and \(T:E\rightarrow E\) be a bounded linear operator.
In fact, in 1958, Browder [10] proved the following.
Theorem B
Let E be a reflexive Banach space. Then a solution u of the equation \(u-Tu=f\) exists for a given point \(f\in E\) and an operator T which is asymptotically bounded (i.e., there exists \(M\geq0\) such that \(\|T^{n}x\|\leq M\) for all \(x\in E\) and \(n\geq1\)) if and only if the sequence \(\{x_{n}\}\) defined by (BP) is bounded for any fixed \(x_{0}\in E\).
But, without any assumption of the reflexivity on E, under a slight sharper condition on T, Browder and Petryshyn proved the following:
Theorem BP
- (1)
If \(f\in R(I-T)\), the sequence \(\{x_{n}\}\) defined by (BP) converges to a solution u of the equation \(u+Tu =f\).
- (2)
If any subsequence \(\{x_{n_{i}}\}\) of the sequence \(\{x_{n}\}\) converges to an element \(y\in E\), then y is a solution of the equation \(y-Ty=f\).
- (3)
If E is a reflexive Banach space and the sequence \(\{x_{n}\}\) is bounded, then the sequence \(\{x_{n}\}\) converges to a solution of the equation \(u+Tu =f\).
Motivated by Theorem BP, we have the concept of the condition (BP) as follows:
Let E be a real normed linear space, C be a nonempty subset of E and \(T:C\rightarrow E\) be a mapping.
Definition 3.1
The pair \((T,C)\) is said to satisfy the condition (BP) if, for any bounded closed subset G of C, \(\{z:z=x-Tx, x\in G\}\) is a closed subset of E.
Let E and F be Banach spaces. Recall that a mapping \(T:E\rightarrow F\) is completely continuous if it is continuous and compact (i.e., C is bounded implies that \(T(C)\) is relatively compact, i.e., \(\overline{T(C)}\) is compact) or a weakly convergent sequence (\(x_{n}\to x\) weakly) implies a strongly convergent sequence (\(Tx_{n}\to Tx\)).
We give some relations between the condition (BP) and a completely continuous mapping as follows.
Proposition 3.1
Let E be a real normed linear space, C be a nonempty subset of E and \(T:C\rightarrow E\) be a completely continuous mapping. Then the pair \((T,C)\) satisfies the condition (BP).
Proof
For any bounded closed subset G of C, we denote \(M=\{z: z=x-Tx, x\in G\}\). For any \(z_{n}\in M\) with \(z_{n}\rightarrow z\), there exists \(x_{n}\in G\) such that \(z_{n}=x_{n}-Tx_{n}\). Since T is completely continuous and the sequence \(\{x_{n}\}\) is bounded, there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\{Tx_{n_{k}}\}\) is convergent. Letting \(Tx_{n_{k}}\rightarrow x_{0}\), it follows that \(x_{n_{k}}=z_{n_{k}}+Tx_{n_{k}}\rightarrow z+x_{0}\) as \(k\rightarrow\infty\). Since G is closed, it follows that \(z+x_{0}\in G\).
Remark 3.1
If the pair \((T,C)\) satisfies the condition (BP), then T is not completely continuous in general.
Example 3.1
4 Strong convergence theorems
Now, we prove strong convergence theorems for asymptotically nonexpansive nonself-mappings with the condition (BP) in real uniformly convex Banach spaces.
Theorem 4.1
If the pair \((T,C)\) satisfies the condition (BP), then the sequence \(\{x_{n}\}\) converges strongly to a fixed point of T.
Proof
Letting \(G=\overline{\{x_{n}\}}\), where \(\overline{\{x_{n}\}}\) denotes the closure of \({\{x_{n}\}}\), since the sequence \(\{x_{n}\}\) is bounded in C by Lemma 2.2(2) and so G is a bounded closed subset of C. Since the pair \((T,C)\) satisfies the condition (BP), it follows that \(M=\{ z=x-Tx:x\in G\}\) is closed. From \(\{x_{n}-Tx_{n}\}\subset M\) and \(x_{n}-Tx_{n}\rightarrow0\) as \(n\rightarrow\infty\) by Lemma 2.3, we know that the zero vector \(0\in M\) and so there exists a \(q\in G\) such that \(q=Tq\). This shows that q is a fixed point of T. Since \(q\in G\), there exists a positive integer \(n_{0}\) such that \(x_{n_{0}}=q\) or there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\rightarrow q\) as \(k\rightarrow\infty\).
If \(x_{n_{0}}=q\), then it follows from Lemma 2.2(1) that \(x_{n}=q\) for all \(n\geq n_{0}\) and so \(x_{n}\rightarrow q\) as \(n\rightarrow\infty\).
If \(x_{n_{k}}\rightarrow q\), then, since \(\lim_{n\rightarrow\infty}\|x_{n}-q\|\) exists, we have \(x_{n}\rightarrow q\) as \(n\rightarrow\infty\). This completes the proof. □
Using Theorem 4.1 and Proposition 3.1, we have the following.
Corollary 4.1
If T is completely continuous, then the sequence \(\{x_{n}\}\) converges strongly to a fixed point of T.
Letting \(\beta_{n}=0\) for all \(n\geq1\) in Theorem 4.1, we have the following.
Theorem 4.2
If the pair \((T,C)\) satisfies the condition (BP), then the sequence \(\{x_{n}\}\) converges strongly to a fixed point of T.
Using Theorem 4.2 and Proposition 3.1, we obtain the following.
Corollary 4.2
If T is completely continuous, then the sequence \(\{x_{n}\}\) converges strongly to a fixed point of T.
5 Examples
Now, we give two examples to illustrate Theorem 4.1 as follows.
Proposition 5.1
([6])
Example 5.1
Example 5.2
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant no. (18-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, this project (WP Guo) was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the third author (YJ Cho) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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