A new hybrid algorithm for a nonexpansive mapping
- Qiao-Li Dong^{1, 2}Email author and
- Yan-Yan Lu^{1}
https://doi.org/10.1186/s13663-015-0285-6
© Dong and Lu; licensee Springer. 2015
Received: 6 November 2014
Accepted: 19 February 2015
Published: 7 March 2015
Abstract
In the paper, we introduce a new hybrid algorithm which is not based on the modification to weak convergence algorithms. The strong convergence theorem of the proposed algorithm is presented. Finally, the numerical experiments suggest that the new algorithm could be faster than Nakajo and Takahashi’s algorithm in J. Math. Anal. Appl. 279:372-379, 2003.
Keywords
MSC
1 Introduction
Let H be a real Hilbert space with the inner product \(\langle\cdot ,\cdot\rangle\) and the norm \(\|\cdot\|\) and C be a nonempty closed convex subset of H. Recall that a mapping \(T : C\rightarrow C\) is said to be nonexpansive if \(\|Tx-Ty\|\leq\|x-y\|\) holds for all \(x, y \in C\). We denote by \(\operatorname{Fix}(T)\) the set of fixed points of T, i.e., \(\operatorname{Fix}(T) = \{x \in C : Tx = x\}\).
Recently, a great deal of literatures on iteration algorithms for approximating fixed points of nonexpansive mappings have been published since they have a variety of applications in inverse problem, image recovery, and signal processing; see [1–7]. Mann’s iteration process [8] is often used to approximate a fixed point of the operators, but it has only weak convergence (see [9] for an example). However, strong convergence is often much more desirable than weak convergence in many problems that arise in infinite dimensional spaces (see [10] and references therein). So, attempts have been made to modify Mann’s iteration process so that strong convergence is guaranteed. Let \(T:C\rightarrow C\) be a nonexpansive mapping such that \(\operatorname{Fix}(T)\neq \emptyset\). Nakajo and Takahashi [11] firstly introduced the following hybrid algorithm.
Algorithm 1
Inspired by the recent work of Malitsky and Semenov [20], we propose the following algorithm.
Algorithm 2
The paper is organized as follows. In the next section, we present some lemmas which will be used in the main results. In Section 3, strong convergence theorem and its proof are given. In the final section, Section 4, some numerical results are provided, which show advantages of our algorithm.
2 Preliminaries
- (1)
⇀ for weak convergence and → for strong convergence.
- (2)
\(\omega_{w}(x_{n}) = \{x : \exists x_{n_{j}}\rightharpoonup x\}\) denotes the weak ω-limit set of \(\{x_{n}\}\).
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 2.1
Lemma 2.2
(Goebel and Kirk [21])
Let C be a closed convex subset of a real Hilbert space H, and let \(T : C \rightarrow C\) be a nonexpansive mapping such that \(\operatorname{Fix}(T)\neq\emptyset\). If a sequence \(\{x_{n}\}\) in C is such that \(x_{n}\rightharpoonup z\) and \(x_{n} - T x_{n} \rightarrow0\), then \(z = T z\).
Lemma 2.3
Lemma 2.4
(Matinez-Yanes and Xu [22])
Lemma 2.5
Proof
3 Algorithm and its convergence
In this section, we present strong convergence theorem and its proof for Algorithm 2.
Theorem 3.1
Let C be a closed convex subset of a Hilbert space H, and let \(T: C \rightarrow C\) be a nonexpansive mapping such that \(\operatorname{Fix}(T)\neq\emptyset \). Assume that \(\{\alpha_{n}\}\subset[0,\sigma]\) holds for some \(\sigma \in [0, \frac{1}{2})\). Then \(\{x_{n}\}\) and \(\{z_{n}\}\) generated by Algorithm 2 converge strongly to \(P_{\operatorname{Fix}(T)}x_{0}\).
Proof
Theorem 3.2
Let C be a closed convex subset of a Hilbert space H, and let \(T: C \rightarrow C\) be a nonexpansive mapping such that \(\operatorname{Fix}(T)\neq\emptyset \). Assume \(\{\alpha_{n}\}\subset[a,b]\) for some \(a,b\in (\frac{1}{2}, 1)\). Then \(\{x_{n}\}\) and \(\{z_{n}\}\) generated by the iteration process (10) strongly converge to \(P_{\operatorname{Fix}(T)}x_{0}\).
4 Numerical experiments
In this section, we firstly present specific expression of \(P_{C_{n}\cap Q_{n}} x_{0}\) in Algorithm 2 and then compare Algorithms 1 and 2 through numerical examples.
He et al. [23] pointed out that it is difficult to realize the hybrid algorithm in actual computing programs because the specific expression of \(P_{C_{n}\cap Q_{n}} x_{0}\) cannot be got, in general. For a special case \(C = H\), where \(C_{n}\) and \(Q_{n}\) are two half-spaces, they obtained the specific expression of \(P_{C_{n}\cap Q_{n}} x_{0}\) and realized Algorithm 1.
Table 1 illustrates that in our examples Algorithm 2 has a competitive performance. We caution, however, that this study is a very preliminary one.
Declarations
Acknowledgements
Supported by the National Natural Science Foundation of China (No. 11201476) and Fundamental Research Funds for the Central Universities (No. 3122013D017), in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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