Inertial algorithm for approximating a common fixed point for a countable family of relatively nonexpansive maps
 C. E. Chidume^{1}Email author,
 S. I. Ikechukwu^{1} and
 A. Adamu^{1}
https://doi.org/10.1186/s1366301806343
© The Author(s) 2018
Received: 9 November 2017
Accepted: 16 February 2018
Published: 12 March 2018
Abstract
In this paper, we study an inertial algorithm for approximating a common fixed point for a countable family of relatively nonexpansive maps in a uniformly convex and uniformly smooth real Banach space. We prove a strong convergence theorem. This theorem is an improvement of the result of Matsushita and Takahashi (J. Approx. Theory 134:257–266, 2005) and the result of Dong et al. (Optim. Lett. 12:87–102, 2018). Finally, we give some applications of our theorem.
Keywords
MSC
1 Introduction
An inertialtype algorithm was first proposed by Polyak [3] as an acceleration process in solving a smooth convex minimisation problem. An inertialtype algorithm is a twostep iterative method in which the next iterate is defined by making use of the previous two iterates. It is well known that incorporating an inertial term in an algorithm speeds up or accelerates the rate of convergence of the sequence generated by the algorithm. Consequently, a lot of research interest is now devoted to the inertialtype algorithm (see e.g. [2, 4, 5] and the references contained in them).
 1.
If E is a reflexive, strictly convex and smooth real Banach space, then J is surjective, injective and singlevalued. If E is uniformly smooth, then J is uniformly continuous on bounded sets.
 2.
In a real Hilbert space H, the duality map J is the identity map on H.
 \((N_{0})\) :

\({ (\x\\y\ )}^{2} \leq\psi(x,y)\leq{ (\x\+\y\ )}^{2}\),
 \((N_{1})\) :

\(\psi(x,y)=\psi(x,z)+\psi(z,y)+2\langle z x, JyJz \rangle\),
 \((N_{2})\) :

\(\psi(x,J^{1}(\alpha Jy+(1\alpha)Jz))\leq\alpha \psi(x,y)+(1\alpha)\psi(x,z)\),
 \((N_{3})\) :

\(\psi(x,y)\leq\x\ \JxJy\+\y\ \xy\\).
A point \(x^{*}\in C\) is called an asymptotic fixed point of T if there exists a sequence \(\{u^{n}\} \subseteq C\) such that \(u^{n} \rightharpoonup x^{*}\) and \(\u^{n}Tu^{n}\ \rightarrow0\), as \(n \rightarrow\infty\) (see e.g. Chang et al. [15]). Here we shall denote the set of asymptotic fixed points of T by \(\widehat{F}(T)\).
Definition 1.1
 (i)
\(\widehat{F}(T)=F(T)\neq\emptyset\), and
 (ii)
\(\psi(p,Tx)\leq\psi(p,x)\), \(\forall x\in C\), \(p\in F(T)\).
Remark 1
Every real Hilbert space is an Opial space, and so if \(\lbrace u^{n} \rbrace\) is a sequence in H such that \(u^{n} \rightharpoonup x^{*}\) and \(\u^{n}Tu^{n}\\rightarrow0\), it is well known that if T is nonexpansive, then \(Tx^{*}=x^{*}\) and \(\widehat{F}(T)=F(T)\). Moreover, since \(\psi(x,y)=\xy\^{2}\), \(\forall x,y \in H\), it follows that T is relatively nonexpansive.
Let \(B:=\lbrace x\in E :\x\=1\rbrace\) be the unit sphere of E. A Banach space E is said to be strictly convex if, for all \(x,y \in B\), \(x \neq y \Rightarrow\frac{\x+y\}{2}<1 \). The space E is said to have the Kadec–Klee property if whenever \(\lbrace u^{n} \rbrace\) is a sequence in E that converges weakly to \(u^{0} \in E\) and \(\u^{n}\\rightarrow\u^{0}\\), as \(n \rightarrow\infty\), then \(\lbrace u^{n}\rbrace\) converges strongly to \(u^{0}\). A space E is said to be uniformly convex if, for each \(\epsilon\in(0,2]\), there exists \(\delta>0\) such that \(\xy\\geq\epsilon\Rightarrow\\frac{x+y}{2}\<1\delta\), \(\forall x, y \in B\). It is well known that a uniformly convex real Banach space is reflexive, strictly convex and has the Kadec–Klee property [17, 18].
Recently, Dong et al. [2] studied the following inertial CQ algorithm for nonexpansive maps in a real Hilbert space. They proved the following theorem:
Theorem 1.2
(Dong et al., [2, Theorem 4.1])
In this paper, motivated by the results of Matsushita and Takahashi [1] and Dong et al. [2], we study an inertial algorithm in a uniformly convex and uniformly smooth real Banach space and prove a strong convergence theorem for the sequence generated by our algorithm. As a consequence of this result, we obtain a strong convergence theorem for approximating a common fixed point for a countable family of relatively nonexpansive maps. Our theorem is an improvement of the results of Dong et al. [2], Matsushita and Takahashi [1], Nakajo and Takahashi [19] and a host of other results.
2 Preliminaries
Lemma 2.1
(Alber [7])
Lemma 2.2
(Kamimura and Takahashi [10])
Let E be a smooth and uniformly convex real Banach space, and let \(\lbrace u^{n}\rbrace\) and \(\lbrace v^{n}\rbrace\) be two sequences of E. If either \(\lbrace u^{n}\rbrace\) or \(\lbrace v^{n}\rbrace\) is bounded and \(\psi(u^{n},v^{n}) \rightarrow0\), as \(n\rightarrow\infty\), then \(\u^{n} v^{n}\ \rightarrow0\), as \(n \rightarrow\infty\).
Remark 2
Using \((N_{3})\), it is easy to see that the converse of Lemma 2.2 is also true whenever \(\lbrace u^{n}\rbrace\) and \(\lbrace v^{n}\rbrace\) are both bounded.
Lemma 2.3
(Matsushita and Takahashi [1])
Let E be a smooth and strictly convex real Banach space, and let C be a nonempty, closed and convex subset of E. Let T be a map from C into itself such that \(F(T)\neq\emptyset\) and \(\psi(y,Tx) \leq\psi(y,x)\), \(\forall (y,x)\in F(T)\times C\). Then \(F(T)\) is closed and convex.
Lemma 2.4
(Kohsaka and Takahashi [20, Theorem 3.3])
3 Main results
We first prove the following lemma which will be central for the proof of our main theorem.
Lemma 3.1
Proof
We partition our proof into four steps.
Step 1. We show that \(\lbrace u^{n}\rbrace\) is well defined and \(F(T)\subseteq C_{n}\), \(\forall n\geq0\).
Step 2. We show that \(\lbrace u^{n}\rbrace\), \(\lbrace v^{n}\rbrace\) and \(\lbrace w^{n}\rbrace\) are bounded.
Using the definition of \(w^{n}\), we have that \(\u^{n}  w^{n}\=\\alpha_{n} (u^{n1}  u^{n})\ \leq\u^{n1}  u^{n}\\rightarrow0\), as \(n\rightarrow\infty\). Since \(\lbrace w^{n}\rbrace\) is bounded, by Remark 2, we have that \(\psi(u^{n},w^{n}) \rightarrow0\), as \(n\rightarrow\infty\). Since \(u^{n+1} \in C_{n} \), it follows that \(0 \leq\psi(u^{n+1},v^{n}) \leq\psi(u^{n+1},w^{n}) \rightarrow0\), as \(n\rightarrow\infty\), which implies that \(\u^{n}  v^{n}\\rightarrow0\), as \(n\rightarrow \infty\); and consequently, \(\lbrace v^{n}\rbrace\) is bounded.
Step 3. We show that \(\w^{n}  Tw^{n}\ \rightarrow0\), as \(n\rightarrow\infty\).
Step 4. We show that \(u^{n}\rightarrow\Pi_{F(T)}u^{0}\).
Since \(\lbrace w^{n} \rbrace\) is bounded, there exists \(\lbrace w^{n_{k}} \rbrace\), a subsequence of \(\lbrace w^{n} \rbrace\), such that \(w^{n_{k}} \rightharpoonup x^{*}\), as \(k \rightarrow\infty\). Using Step 3, we obtain that \(\w^{n_{k}}  Tw^{n_{k}}\ \rightarrow0\), as \(k\rightarrow \infty\). Since our map is relatively nonexpansive, we have that \(x^{*} \in F(T)\). Thus, it follows from Step 2 that there exists \(\lbrace u^{n_{k}} \rbrace\), a subsequence of \(\lbrace u^{n} \rbrace\), such that \(u^{n_{k}} \rightharpoonup x^{*}\), as \(k \rightarrow\infty\). We now show that \(x^{*}=\Pi_{F(T)}u^{0}\). Set \(v=\Pi_{F(T)}u^{0}\).
Using Lemma 3.1, we now prove our main theorem of this paper.
Theorem 3.2
Proof
From Lemma 2.4, T is relatively nonexpansive and \(F(T)=\bigcap _{i=1}^{\infty}F(T_{i})\). The conclusion follows from Lemma 3.1. □
Corollary 3.3
Corollary 3.4
4 Conclusion

The algorithms studied in Matsushita and Takahashi [1] and Dong et al. [2] require at each step of the iteration process the computation of two subsets \(C_{n}\) and \(Q_{n}\) of C; their intersection \(C_{n} \cap Q_{n}\), and the projection of the initial vector onto this intersection. In our iteration process, the subset \(Q _{n}\) has been dispensed with. Furthermore, the sequences \(\lbrace\alpha_{n}\rbrace\) and \(\lbrace \beta_{n}\rbrace\) used in the algorithms of Matsushita and Takahashi [1] and Dong et al. [2], which are also to be computed at each step of the iteration process, have been replaced by a fixed constant β in our algorithm. This β is to be computed once and used at each step of the iteration process. Consequently, our algorithm reduces computational cost.

In Matsushita and Takahashi [1], the authors proved a strong convergence theorem for a relatively nonexpansive map \(T : C \rightarrow C\). In our Theorem 3.2, a strong convergence theorem is proved for a countable family of relatively nonexpansive maps \(T_{i} : E \rightarrow E\), \(i\in\mathbb{N}\). Furthermore, unlike the algorithm of Matsushita and Takahashi, our algorithm has an inertial term which is known to improve the speed of convergence over algorithms without an inertial term (see e.g. [2–5, 21] and the references contained in them).

In Dong et al. [2, Theorem 4.1], the authors proved a strong convergence theorem in a real Hilbert space for one nonexpansive map. Our theorem is proved in the much more general uniformly convex and uniformly smooth real Banach spaces and for a countable family of relatively nonexpansive maps.
Declarations
Acknowledgements
The authors thank the African Capacity Building Foundation (ACBF) for sponsoring this research work.
Funding
Research supported from ACBF Research Grant Funds to AUST.
Authors’ contributions
CEC formulated the problem and suggested the method of proof of the Theorem to SII and AA. The computations using the method suggested by CEC were carried out by SII and AA. The analysis of the computations to arrive at the proof of the Theorem was done jointly by CEC, SII, and AA. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no conflict of interest.
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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