Strongly relatively nonexpansive sequences generated by firmly nonexpansivelike mappings
 Koji Aoyama^{1}Email author and
 Fumiaki Kohsaka^{2}
https://doi.org/10.1186/16871812201495
© Aoyama and Kohsaka; licensee Springer. 2014
Received: 28 June 2013
Accepted: 21 March 2014
Published: 9 April 2014
Abstract
We show that a strongly relatively nonexpansive sequence of mappings can be constructed from a given sequence of firmly nonexpansivelike mappings in a Banach space. Using this result, we study the problem of approximating common fixed points of such a sequence of mappings.
MSC:47H09, 47H05, 65J15.
Keywords
1 Introduction
The aim of the present paper is twofold. Firstly, we construct a strongly relatively nonexpansive sequence from a given sequence of firmly nonexpansivelike mappings with a common fixed point in Banach spaces. Secondly, we obtain two convergence theorems for firmly nonexpansivelike mappings in Banach spaces and discuss their applications.
The class of firmly nonexpansivelike mappings (or mappings of type (P)) introduced in [1] plays an important role in nonlinear analysis and optimization. In fact, the fixed point theory for such mappings can be applied to several nonlinear problems such as zero point problems for monotone operators, convex feasibility problems, convex minimization problems, equilibrium problems, and so on; see [1–3] and Section 5 for more details.
Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space X, J the normalized duality mapping of X into ${X}^{\ast}$, and $T:C\to X$ a firmly nonexpansivelike mapping; see (2.16). The set of all fixed points of T is denoted by $\mathrm{F}(T)$. It is known [[1], Theorem 7.4] that if C is bounded, then $\mathrm{F}(T)$ is nonempty. Martinet’s theorem [[4], Théorème 1] ensures that if X is a Hilbert space and C is bounded, then the sequence $\{{T}^{n}x\}$ converges weakly to an element of $\mathrm{F}(T)$ for each $x\in C$. However, we do not know whether Martinet’s theorem holds for firmly nonexpansivelike mappings in Banach spaces.
On the other hand, using the metric projections in Banach spaces, Kimura and Nakajo [[5], Theorems 6 and 7] recently obtained generalizations of the results due to Crombez [[6], Theorem 3] and Brègman [[7], Theorem 1].
for all $n\in \mathbb{N}$, respectively.
This paper is organized as follows: In Section 2, we give some definitions and state some known results. In Section 3, we obtain two lemmas for a single firmly nonexpansivelike mapping. In Section 4, we construct strongly relatively nonexpansive sequences of mappings from a given sequence of firmly nonexpansivelike mappings. Using these results, we deduce two convergence theorems. In Section 5, we discuss some applications of our results.
2 Preliminaries
Throughout the present paper, we denote by ℕ the set of all positive integers, ℝ the set of all real numbers, X a real Banach space with dual ${X}^{\ast}$, $\parallel \cdot \parallel $ the norms of X and ${X}^{\ast}$, $\u3008x,{x}^{\ast}\u3009$ the value of ${x}^{\ast}\in {X}^{\ast}$ at $x\in X$, ${x}_{n}\to x$ strong convergence of a sequence $\{{x}_{n}\}$ of X to $x\in X$, ${x}_{n}\rightharpoonup x$ weak convergence of a sequence $\{{x}_{n}\}$ of X to $x\in X$, ${S}_{X}$ the unit sphere of X, and ${B}_{X}$ the closed unit ball of X.
for all $\epsilon \in [0,2]$. The space X is said to be 2uniformly convex if there exists $c>0$ such that ${\delta}_{X}(\epsilon )\ge c{\epsilon}^{2}$ for all $\epsilon \in [0,2]$. It is obvious that every 2uniformly convex Banach space is uniformly convex. It is known that all Hilbert spaces are uniformly smooth and 2uniformly convex. It is also known that all the Lebesgue spaces ${L}^{p}$ are uniformly smooth and 2uniformly convex whenever $1<p\le 2$; see [[8], pp.198203]. For a smooth Banach space, J is said to be weakly sequentially continuous if $\{J{x}_{n}\}$ converges weakly^{∗} to Jx whenever $\{{x}_{n}\}$ is a sequence of X such that ${x}_{n}\rightharpoonup x\in X$. We know the following fundamental result.
for all $x,y\in X$.
The minimum value of the set of all $\mu \ge 1$ satisfying (2.4) for all $x,y\in X$ is denoted by ${\mu}_{X}$ and is called the 2uniform convexity constant of X; see [9]. It is obvious that ${\mu}_{X}=1$ whenever X is a Hilbert space.
In what follows throughout this section, we assume the following:

X is a smooth, strictly convex, and reflexive Banach space;

C is a nonempty closed convex subset of X.
for all $x,y,z\in X$. Using Lemma 2.1, we can show the following lemma.
for all $x,y\in X$.
Therefore, we obtain ${(\parallel xy\parallel /{\mu}_{X})}^{2}\le \varphi (x,y)$ as desired. □
see [[13], Remark 7.3] and [[14], Proposition 4].
for all $x,y\in C$; see also [2, 3]. The set of all fixed points of T is denoted by $\mathrm{F}(T)$. If X is a Hilbert space, then T is firmly nonexpansivelike if and only if it is firmly nonexpansive, i.e., ${\parallel TxTy\parallel}^{2}\le \u3008TxTy,xy\u3009$ for all $x,y\in C$. It is known [1] that the following hold:

the metric projection ${P}_{C}$ of X onto C is a firmly nonexpansivelike mapping and $\mathrm{F}({P}_{C})=C$;

if $A:X\to {2}^{{X}^{\ast}}$ is maximal monotone and $\lambda >0$, then the resolvent ${K}_{\lambda}:X\to X$ of A defined by ${K}_{\lambda}={(I+\lambda {J}^{1}A)}^{1}$ is a firmly nonexpansivelike mapping and $\mathrm{F}({K}_{\lambda})={A}^{1}0$.
Let $T:C\to X$ be a mapping. A point $p\in C$ is said to be an asymptotic fixed point of T if there exists a sequence $\{{x}_{n}\}$ of C such that ${x}_{n}\rightharpoonup p$ and ${x}_{n}T{x}_{n}\to 0$; see [15, 16]. The set of all asymptotic fixed points of T is denoted by $\stackrel{\u02c6}{\mathrm{F}}(T)$. The mapping T is said to be of type (r) if $\mathrm{F}(T)$ is nonempty and $\varphi (u,Tx)\le \varphi (u,x)$ for all $u\in \mathrm{F}(T)$ and $x\in C$. It is known that if T is of type (r), then $\mathrm{F}(T)$ is closed and convex; see [[16], Proposition 2.4]. The mapping T is said to be of type (sr) if T is of type (r) and $\varphi (T{z}_{n},{z}_{n})\to 0$ whenever $\{{z}_{n}\}$ is a bounded sequence of C such that $\varphi (u,{z}_{n})\varphi (u,T{z}_{n})\to 0$ for some $u\in \mathrm{F}(T)$; see [17]. We know the following results:
Lemma 2.3 ([[3], Lemma 2.2])
If $T:C\to X$ is a firmly nonexpansivelike mapping, then $\mathrm{F}(T)$ is a closed convex subset of X and $\stackrel{\u02c6}{\mathrm{F}}(T)=\mathrm{F}(T)$.
Lemma 2.4 ([[17], Lemmas 3.2 and 3.3])
Suppose that X is uniformly convex. If $S:X\to X$ and $T:C\to X$ are mappings of type (r) such that $\mathrm{F}(S)\cap \mathrm{F}(T)$ is nonempty and S or T is of type (sr), then $ST:C\to X$ is of type (r) and $\mathrm{F}(ST)=\mathrm{F}(S)\cap \mathrm{F}(T)$. Further, if both S and T are of type (sr), then so is ST.
Let $\{{T}_{n}\}$ be a sequence of mappings of C into X. The set of all common fixed points of $\{{T}_{n}\}$ is denoted by $\mathrm{F}(\{{T}_{n}\})$. The sequence $\{{T}_{n}\}$ is said to be of type (sr) (or strongly relatively nonexpansive) if $\mathrm{F}(\{{T}_{n}\})$ is nonempty, each ${T}_{n}$ is of type (r), and $\varphi ({T}_{n}{z}_{n},{z}_{n})\to 0$ whenever $\{{z}_{n}\}$ is a bounded sequence of C such that $\varphi (u,{z}_{n})\varphi (u,{T}_{n}{z}_{n})\to 0$ for some $u\in \mathrm{F}(\{{T}_{n}\})$; see [18]. The sequence $\{{T}_{n}\}$ is said to satisfy the condition (Z) if every weak subsequential limit of $\{{x}_{n}\}$ belongs to $\mathrm{F}(\{{T}_{n}\})$ whenever $\{{x}_{n}\}$ is a bounded sequence of C such that ${x}_{n}{T}_{n}{x}_{n}\to 0$; see [18].
Remark 2.5 For a mapping T of C into X, the following hold: T is of type (sr) if and only if $\{T,T,\dots \}$ is of type (sr); $\stackrel{\u02c6}{\mathrm{F}}(T)=\mathrm{F}(T)$ if and only if $\{T,T,\dots \}$ satisfies the condition (Z).
We know the following fundamental results; see [[18], Theorem 3.4] for (i) and [[19], Propositions 3 and 6] for (ii).
 (i)
$\{{S}_{n}{T}_{n}\}$ is of type (sr);
 (ii)
if X is uniformly smooth and both $\{{S}_{n}\}$ and $\{{T}_{n}\}$ satisfy the condition (Z), then so does $\{{S}_{n}{T}_{n}\}$.
We know the following result; see [[18], Theorem 4.1] for (i) and [[20], Theorem 4.1] for (ii).
 (i)
if ${T}_{n}(C)\subset C$ for all $n\in \mathbb{N}$ and J is weakly sequentially continuous, then the sequence $\{{x}_{n}\}$ defined by ${x}_{1}\in C$ and ${x}_{n+1}={T}_{n}{x}_{n}$ for all $n\in \mathbb{N}$ converges weakly to the strong limit of $\{{Q}_{F}{x}_{n}\}$;
 (ii)
if u is an element of X and $\{{\alpha}_{n}\}$ is a sequence of $[0,1]$ such that ${\alpha}_{n}>0$ for all $n\in \mathbb{N}$, ${\alpha}_{n}\to 0$, and ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, then the sequence $\{{y}_{n}\}$ defined by ${y}_{1}\in C$ and ${y}_{n+1}={Q}_{C}{J}^{1}({\alpha}_{n}Ju+(1{\alpha}_{n})J{T}_{n}{y}_{n})$ for all $n\in \mathbb{N}$ converges strongly to ${Q}_{F}u$.
3 Lemmas
Throughout this section, we assume the following:

C is a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space X;

T is a firmly nonexpansivelike mapping of C into X;

S is a mapping of C into X defined by $S={J}^{1}(J\beta J(IT))$, where $\beta >0$ and I denotes the identity mapping on C.
 (i)
$\mathrm{F}(S)=\mathrm{F}(T)$ and $\mathrm{F}({Q}_{C}S)=\mathrm{F}({P}_{C}T)$;
 (ii)
if $\mathrm{F}(T)$ is nonempty, then $\mathrm{F}({P}_{C}T)=\mathrm{F}(T)$.
Thus we have $\mathrm{F}({Q}_{C}S)=\mathrm{F}({P}_{C}T)$.
By (3.2) and (3.3), we obtain ${\parallel Tvv\parallel}^{2}\le 0$. Thus we know that $v\in \mathrm{F}(T)$. □
for all $u\in \mathrm{F}(S)$ and $x\in C$.
By (3.8) and (3.9), we obtain the desired inequality. □
4 Construction of strongly relatively nonexpansive sequences
Throughout this section, we assume the following:

C is a nonempty closed convex subset of a smooth and 2uniformly convex Banach space X;

$\{{T}_{n}\}$ is a sequence of firmly nonexpansivelike mappings of C into X such that $F=\mathrm{F}(\{{T}_{n}\})$ is nonempty;

$\{{S}_{n}\}$ is a sequence of mappings of C into X defined by${S}_{n}={J}^{1}(J{\beta}_{n}J(I{T}_{n}))$(4.1)
for all $n\in \mathbb{N}$, where $\{{\beta}_{n}\}$ is a sequence of real numbers such that $0<{inf}_{n}{\beta}_{n}$ and ${sup}_{n}{\beta}_{n}<2{({\mu}_{X})}^{2}$ and I denotes the identity mapping on C.
 (i)
$\mathrm{F}(\{{S}_{n}\})=F$ and $\{{S}_{n}\}$ is of type (sr);
 (ii)
if X is uniformly smooth and $\{{T}_{n}\}$ satisfies the condition (Z), then so does $\{{S}_{n}\}$.
Thus it follows from ${sup}_{n}{\beta}_{n}<2{({\mu}_{X})}^{2}$ that $\parallel {S}_{n}{z}_{n}{z}_{n}\parallel \to 0$. Consequently, we have $\varphi ({S}_{n}{z}_{n},{z}_{n})\to 0$ and hence $\{{S}_{n}\}$ is of type (sr).
By assumption, we know that $p\in F=\mathrm{F}(\{{S}_{n}\})$. Therefore, $\{{S}_{n}\}$ satisfies the condition (Z). □
By Lemma 2.3, Remark 2.5, and Theorem 4.1, we obtain the following.
 (i)
$\mathrm{F}(S)=\mathrm{F}(T)$ and S is of type (sr);
 (ii)
if X is uniformly smooth, then $\stackrel{\u02c6}{\mathrm{F}}(S)=\mathrm{F}(S)$.
We next show one of our main results in the present paper.
 (i)
$\mathrm{F}(\{{U}_{n}\})=F$ and $\{{U}_{n}\}$ is of type (sr);
 (ii)
if X is uniformly smooth and $\{{T}_{n}\}$ satisfies the condition (Z), then so does $\{{U}_{n}\}$.
the part (i) of Lemma 2.6 implies that $\{{U}_{n}\}$ is of type (sr).
We finally show (ii). Suppose that X is uniformly smooth and $\{{T}_{n}\}$ satisfies the condition (Z). Since C is weakly closed, we can easily see that $\stackrel{\u02c6}{\mathrm{F}}({Q}_{C})=\mathrm{F}({Q}_{C})=C$. This implies that $\{{Q}_{C},{Q}_{C},\dots \}$ satisfies the condition (Z). By (ii) of Theorem 4.1, we know that $\{{S}_{n}\}$ satisfies the condition (Z). Thus (ii) of Lemma 2.6 implies the conclusion. □
By Lemma 2.3, Remark 2.5, and Theorem 4.3, we obtain the following.
 (i)
$\mathrm{F}(U)=\mathrm{F}(T)$ and U is of type (sr);
 (ii)
if X is uniformly smooth, then $\stackrel{\u02c6}{\mathrm{F}}(U)=\mathrm{F}(U)$.
As a direct consequence of (i) of Theorem 2.7 and Theorem 4.3, we obtain the following result.
for all $n\in \mathbb{N}$. If J is weakly sequentially continuous, then $\{{x}_{n}\}$ converges weakly to the strong limit of $\{{Q}_{F}{x}_{n}\}$.
As a direct consequence of (ii) of Theorem 2.7 and Theorem 4.1, we obtain the following result.
for all $n\in \mathbb{N}$. Then $\{{y}_{n}\}$ converges strongly to ${Q}_{F}u$.
By Lemma 2.3, Theorem 4.5, and Theorem 4.6, we obtain the following corollary for a single firmly nonexpansivelike mapping.
 (i)
if J is weakly sequentially continuous, then the sequence $\{{x}_{n}\}$ defined by ${x}_{1}\in C$ and (1.1) for all $n\in \mathbb{N}$ converges weakly to the strong limit of $\{{Q}_{\mathrm{F}(T)}{x}_{n}\}$;
 (ii)
if $\{{\alpha}_{n}\}$ is a sequence of $[0,1]$ such that ${\alpha}_{n}>0$ for all $n\in \mathbb{N}$, ${\alpha}_{n}\to 0$, and ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, then the sequence $\{{y}_{n}\}$ defined by ${y}_{1}\in C$ and (1.2) for all $n\in \mathbb{N}$ converges strongly to ${Q}_{\mathrm{F}(T)}u$.
Remark 4.8 Since ${\mu}_{X}=1$ and J is the identity mapping on C in the case when X is a Hilbert space, the part (i) of Corollary 4.7 is a generalization of Martinet’s theorem [[4], Théorème 1].
5 Applications
Using Theorem 4.5, we first study the problem of approximating zero points of maximal monotone operators.
for all $n\in \mathbb{N}$. If J is weakly sequentially continuous, then $\{{x}_{n}\}$ converges weakly to the strong limit of $\{{Q}_{F}{x}_{n}\}$.
Proof It is well known that each ${K}_{{\lambda}_{n}}$ is a single valued mapping of X into itself and $\mathrm{F}({K}_{{\lambda}_{n}})=F$; see [21, 22]. We also know that each ${K}_{{\lambda}_{n}}$ is firmly nonexpansivelike and $\{{K}_{{\lambda}_{n}}\}$ satisfies the condition (Z); see [1, 3]. Therefore, Theorem 4.5 implies the conclusion. □
Remark 5.2 Corollary 5.1 is a generalization of Rockafellar’s weak convergence theorem [23] for the proximal point algorithm in Hilbert spaces.
for all $x\in X$.
for all $n\in \mathbb{N}$. If J is weakly sequentially continuous, then $\{{x}_{n}\}$ converges weakly to the strong limit of $\{{Q}_{F}{x}_{n}\}$.
for all $\lambda >0$ and $x\in X$, where I denotes the identity mapping on X. Therefore, the result follows from Corollary 5.1. □
Using Theorem 4.6, we can similarly show the following corollary.
for all $n\in \mathbb{N}$, where $\{{\alpha}_{n}\}$ is a sequence of $[0,1]$ such that ${\alpha}_{n}>0$ for all $n\in \mathbb{N}$, ${\alpha}_{n}\to 0$, and ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$. Then $\{{y}_{n}\}$ converges strongly to ${Q}_{F}u$.
Using the results obtained in Section 4 and (i) of Theorem 2.7, we next study the problem of approximating common points of a given family of closed convex sets.
for all $n\in \mathbb{N}$. If J is weakly sequentially continuous, then $\{{x}_{n}\}$ converges weakly to the strong limit of $\{{Q}_{F}{x}_{n}\}$.
for all $n\in \mathbb{N}$. Note that ${x}_{n+1}={U}_{n}{V}_{n}{W}_{n}{x}_{n}$ for all $n\in \mathbb{N}$, ${P}_{{C}_{k}}$ is firmly nonexpansivelike, and $\stackrel{\u02c6}{\mathrm{F}}({P}_{{C}_{k}})=\mathrm{F}({P}_{{C}_{k}})={C}_{k}$ for all $k\in \{1,2,3\}$. By Theorem 4.1 and Corollary 4.2, we know that the following hold:

$\mathrm{F}(\{{U}_{n}\})={C}_{1}$, $\mathrm{F}(\{{V}_{n}\})={C}_{2}$, and $\mathrm{F}(\{{W}_{n}\})={C}_{3}$;

${U}_{n}$, ${V}_{n}$, and ${W}_{n}$ are of type (sr) for all $n\in \mathbb{N}$;

$\{{U}_{n}\}$, $\{{V}_{n}\}$, and $\{{W}_{n}\}$ are of type (sr);

$\{{U}_{n}\}$, $\{{V}_{n}\}$, and $\{{W}_{n}\}$ satisfy the condition (Z).
Lemmas 2.4 and 2.6 ensure that the following hold:

$\mathrm{F}(\{{U}_{n}{V}_{n}\})=\mathrm{F}(\{{U}_{n}\})\cap \mathrm{F}(\{{V}_{n}\})={C}_{1}\cap {C}_{2}$;

each ${U}_{n}{V}_{n}$ is of type (sr);

$\{{U}_{n}{V}_{n}\}$ is of type (sr) and satisfies the condition (Z).
Lemmas 2.4 and 2.6 also ensure that $\mathrm{F}(\{{U}_{n}{V}_{n}{W}_{n}\})=\mathrm{F}(\{{U}_{n}{V}_{n}\})\cap \mathrm{F}(\{{W}_{n}\})=F$, $\{{U}_{n}{V}_{n}{W}_{n}\}$ is of type (sr), and $\{{U}_{n}{V}_{n}{W}_{n}\}$ satisfies the condition (Z). Therefore, (i) of Theorem 2.7 implies the conclusion. □
Using the results obtained in Section 4 and (ii) of Theorem 2.7, we can similarly show the following result.
for all $n\in \mathbb{N}$, where $\{{\alpha}_{n}\}$ is a sequence of $[0,1]$ such that ${\alpha}_{n}>0$ for all $n\in \mathbb{N}$, ${\alpha}_{n}\to 0$, and ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$. Then $\{{y}_{n}\}$ converges strongly to ${Q}_{F}u$.
Declarations
Acknowledgements
The authors would like to express their sincere appreciation to the anonymous referees for their helpful comments on the original version of the manuscript.
Authors’ Affiliations
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