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Uniform mean convergence theorems for hybrid mappings in Hilbert spaces
 Koji Aoyama^{1} and
 Fumiaki Kohsaka^{2}Email author
https://doi.org/10.1186/168718122012193
© Aoyama and Kohsaka; licensee Springer 2012
 Received: 29 June 2012
 Accepted: 15 October 2012
 Published: 29 October 2012
Abstract
Using the notion of sequences of means on the Banach space of all bounded real sequences, we prove mean and uniform mean convergence theorems for pointwise convergent sequences of hybrid mappings in Hilbert spaces.
MSC:47H25, 47H09, 47H10, 40H05.
Keywords
 ergodic theorem
 hybrid mapping
 nonexpansive mapping
 nonspreading mapping
 uniform mean convergence theorem
1 Introduction
Using the notion of asymptotically invariant sequences of means on ${l}^{\mathrm{\infty}}$, we obtain a mean convergence theorem for pointwise convergent sequences of hybrid mappings in Hilbert spaces. By assuming the strong regularity on the sequences of means, we also obtain a uniform mean convergence theorem.
In 1975, Baillon [1] established a nonlinear ergodic theorem for nonexpansive mappings in Hilbert spaces. Several results related to Baillon’s ergodic theorem have been obtained since then; see, for instance, [2–8] and the references therein. Especially, using the notion of asymptotically invariant nets of means on semitopological semigroups, Hirano, Kido, and Takahashi [4] and Lau, Shioji, and Takahashi [5] generalized Baillon’s ergodic theorem to commutative and noncommutative semigroups of nonexpansive mappings in Banach spaces, respectively.
On the other hand, Akatsuka, Aoyama, and Takahashi [9] obtained another generalization of Baillon’s ergodic theorem for pointwise convergent sequences of nonexpansive mappings in Hilbert spaces. Their result was applied to the problem of approximating common fixed points of countable families of nonexpansive mappings. Recently, the authors [10] generalized some results in [9] for pointwise convergent sequences of hybrid mappings in the sense of [11].
The aim of the present paper is to obtain further generalizations of the results in [9, 10] by using a sequence $\{{\mu}_{n}\}$ of means on ${l}^{\mathrm{\infty}}$. In particular, by assuming the strong regularity on $\{{\mu}_{n}\}$, we prove a uniform mean convergence theorem (Theorem 3.5) for pointwise convergent sequences of hybrid mappings in Hilbert spaces.
Our paper is organized as follows. In Section 2, we recall some definitions and some preliminary results. In Section 3, we prove mean convergence theorems by using sequences of means on ${l}^{\mathrm{\infty}}$; see Theorems 3.4 and 3.5. In Section 4, we obtain some consequences of Theorem 3.5; see Theorems 4.1, 4.2, and 4.3. In Section 5, we give two applications of Theorem 4.3.
2 Preliminaries
Throughout the present paper, every linear space is real. We denote the sets of all nonnegative integers and all real numbers by ℕ and ℝ, respectively. For a Banach space X, the conjugate space of X is denoted by ${X}^{\ast}$. We denote the norms of X and ${X}^{\ast}$ by $\parallel \cdot \parallel $. For a sequence $\{{x}_{n}\}$ of a Banach space X and $x\in X$, strong and weak convergence of $\{{x}_{n}\}$ to x are denoted by ${x}_{n}\to x$ and ${x}_{n}\rightharpoonup x$, respectively. For a sequence $\{{x}_{n}^{\ast}\}$ of ${X}^{\ast}$ and ${x}^{\ast}\in {X}^{\ast}$, weak^{∗} convergence of $\{{x}_{n}^{\ast}\}$ to ${x}^{\ast}$ is also denoted by . The inner product of a Hilbert space H is denoted by $\u3008\cdot ,\cdot \u3009$. For a subset A of a Hilbert space H, the closure of the convex hull of A is denoted by $\overline{co}\phantom{\rule{0.2em}{0ex}}A$.
Since $1\lambda <0$, we obtain $Tx=x$.
We denote the set of all λhybrid mappings of C into H by ${H}_{\lambda}(C,H)$. We also denote by ${H}_{\lambda}(C)$ the set of all λhybrid mappings of C into itself. The set of all fixed points of a mapping $T:C\to H$ is denoted by $F(T)$. A mapping $T:C\to H$ is said to be quasinonexpansive if $F(T)$ is nonempty and $\parallel uTx\parallel \le \parallel ux\parallel $ for all $u\in F(T)$ and $x\in C$. It is well known that $F(T)$ is closed and convex if $T:C\to H$ is quasinonexpansive and C is closed and convex. It is obvious that if $T\in {H}_{\lambda}(C,H)$ for some $\lambda \in \mathbb{R}$ and $F(T)$ is nonempty, then T is quasinonexpansive. We denote the identity mapping on C by I or ${T}^{0}$, where $T:C\to H$ is a mapping.
We know the following lemma.
Lemma 2.1 ([[14], Lemma 3.2])
Let S be a nonempty closed convex subset of a Hilbert space H and $\{{x}_{n}\}$ a sequence of H such that $\parallel u{x}_{n+1}\parallel \le \parallel u{x}_{n}\parallel $ for all $u\in S$ and $n\in \mathbb{N}$. Then $\{{P}_{S}{x}_{n}\}$ converges strongly.
for all $f\in {l}^{\mathrm{\infty}}$, where $(f(k+1))=(f(1),f(2),\dots )$; see [[8], Theorem 1.4.3]. Such a mean μ is called a Banach limit. If μ is a Banach limit and $f\in {l}^{\mathrm{\infty}}$ is convergent, then $\mu (f)={lim}_{k}f(k)$.
Some examples of strongly regular sequences of means on ${l}^{\mathrm{\infty}}$ are shown in Sections 4 and 5. See [15] on asymptotically invariant nets of means and [2, 4–8] on the nonlinear ergodic theory for nonexpansive mappings with asymptotically invariant nets of means. The following lemma is well known.
Lemma 2.2 Let $\{{\mu}_{n}\}$ be an asymptotically invariant sequence of means on ${l}^{\mathrm{\infty}}$ and $\{{\mu}_{{n}_{\alpha}}\}$ a subnet of $\{{\mu}_{n}\}$ such that . Then μ is a Banach limit.
For the sake of completeness, we give the proof.
Proof Since the norm of ${({l}^{\mathrm{\infty}})}^{\ast}$ is weakly^{∗} lower semicontinuous and for each $n\in \mathbb{N}$, we have . On the other hand, since and ${\mu}_{n}(e)=1$ for each $n\in \mathbb{N}$, we obtain $\mu (e)={lim}_{\alpha}{\mu}_{{n}_{\alpha}}(e)=1$. This implies that $1=\mu (e)\le \parallel \mu \parallel $. Hence, μ is a mean on ${l}^{\mathrm{\infty}}$.
Thus, μ is a Banach limit. □
for all $y\in H$. In this case, it is known that ; see [7, 8] for more details. It is easy to see that if μ is a Banach limit and $\{{x}_{n}\}$ is a sequence of H which converges weakly to $p\in H$, then $G(\{{x}_{k}\},\mu )=p$. We need the following lemma in the proof of Theorem 3.1.
Lemma 2.3 Let H be a Hilbert space, $\{{x}_{n}\}$ a bounded sequence of H, $\{{y}_{n}\}$ a strongly convergent sequence of H, and $({\beta}_{n})$ a convergent sequence of real numbers. Then ${[\mu ]}_{n}({\beta}_{n}\u3008{x}_{n}{x}_{n+1},{y}_{n}\u3009)=0$ for each Banach limit μ.
tend to 0, we have ${[\mu ]}_{n}({\beta}_{n}\u3008{x}_{n}{x}_{n+1},{y}_{n}\u3009)=\beta {[\mu ]}_{n}\u3008{x}_{n}{x}_{n+1},y\u3009=0$. □
3 Mean convergence theorems
In this section, we show mean convergence theorems for a pointwise convergent sequence of mappings in ${\bigcup}_{\lambda \in \mathbb{R}}{H}_{\lambda}(C)$.
Throughout this section, we suppose the following conditions:

C is a nonempty closed convex subset of a Hilbert space H;

$({\lambda}_{n})$ is a sequence of real numbers which tends to $\lambda \in \mathbb{R}$;

$\{{T}_{n}\}$ is a sequence of mappings such that ${T}_{n}\in {H}_{{\lambda}_{n}}(C)$ for all $n\in \mathbb{N}$ and $\{{T}_{n}x\}$ converges strongly for all $x\in C$;

T is a mapping of C into itself defined by $Tx={lim}_{n}{T}_{n}x$ for all $x\in C$;

$\{{x}_{n}\}$ is a sequence of C defined by ${x}_{0}\in C$ and ${x}_{n+1}={T}_{n}{x}_{n}$ for all $n\in \mathbb{N}$.
Motivated by [7–10, 12], we first show the following fundamental theorem.
Theorem 3.1 If $\{{x}_{n}\}$ is bounded, then $G(\{{x}_{k}\},\mu )$ is a fixed point of T for each Banach limit μ.
It follows from (3.4) and (3.5) that $0\le {\parallel Tzz\parallel}^{2}$. Therefore, z is a fixed point of T. □
Using Lemma 2.1 and Theorem 3.1, we next show the following theorem.
for each Banach limit μ.
for all $u\in F(T)$ and $n\in \mathbb{N}$. It also follows from (3.7) that $\{{x}_{n}\}$ is bounded. According to Theorem 3.1, we know that $G(\{{x}_{k}\},\mu )$ is a fixed point of T. Using Lemma 2.1 and (3.7), we also know that $\{P{x}_{n}\}$ converges strongly to some $w\in F(T)$.
Therefore, $z=w$. □
As a direct consequence of Theorems 3.1 and 3.2, we can obtain the following corollary for a single hybrid mapping.
 (i)
if $\{{S}^{n}x\}$ is bounded, then $F(S)$ is nonempty and $G(\{{S}^{k}x\},\mu )$ is a fixed point of S for each Banach limit μ;
 (ii)if $F(S)$ is nonempty, then $\{{S}^{n}x\}$ is bounded, $\{{P}_{F(S)}{S}^{n}x\}$ is strongly convergent, and$G(\left\{{S}^{k}x\right\},\mu )=\underset{n\to \mathrm{\infty}}{lim}{P}_{F(S)}{S}^{n}x$(3.13)
for each Banach limit μ.
Using the notion of an asymptotically invariant sequence of means on ${l}^{\mathrm{\infty}}$, we next show the following mean convergence theorem.
converges weakly to the strong limit of $\{{P}_{F(T)}{x}_{n}\}$.
Proof By Theorem 3.2, we know that $\{{x}_{n}\}$ is bounded and $\{{P}_{F(T)}{x}_{n}\}$ converges strongly to some $w\in F(T)$.
Let $\{{z}_{n}\}$ be the sequence defined by ${z}_{n}={G}_{{\mu}_{n}}(\{{x}_{k}\})$ for all $n\in \mathbb{N}$. Since for all $n\in \mathbb{N}$, the sequence $\{{z}_{n}\}$ is bounded. Let u be any weak subsequential limit of $\{{z}_{n}\}$. Then we have a subsequence $\{{z}_{{n}_{i}}\}$ of $\{{z}_{n}\}$ such that ${z}_{{n}_{i}}\rightharpoonup u$. It follows from $\parallel {\mu}_{{n}_{i}}\parallel =1$ that there exists a subnet $\{{\mu}_{{n}_{{i}_{\alpha}}}\}$ of $\{{\mu}_{{n}_{i}}\}$ such that . Since $\{{\mu}_{{n}_{i}}\}$ is asymptotically invariant, Lemma 2.2 implies that μ is a Banach limit.
for all $y\in H$. Thus, $\{{z}_{{n}_{{i}_{\alpha}}}\}$ converges weakly to w. On the other hand, since ${z}_{{n}_{i}}\rightharpoonup u$ and $\{{z}_{{n}_{{i}_{\alpha}}}\}$ is a subnet of $\{{z}_{{n}_{i}}\}$, we know that ${z}_{{n}_{{i}_{\alpha}}}\rightharpoonup u$. Accordingly, we have $u=w$. Thus, $\{{z}_{n}\}$ converges weakly to $w={lim}_{n}{P}_{F(T)}{x}_{n}$. □
As in the proof of [[5], the corollary of Theorem 2], we can also show the following uniform mean convergence theorem in the case when the strong regularity of $\{{\mu}_{n}\}$ is assumed.
converges weakly to the strong limit of $\{{P}_{F(T)}{x}_{n}\}$ as $n\to \mathrm{\infty}$ uniformly in $p\in \mathbb{N}$.
Proof Set ${z}_{n,p}={G}_{{r}_{p}^{\ast}{\mu}_{n}}(\{{x}_{k}\})$ for all $n,p\in \mathbb{N}$. It is easy to see that ${r}_{p}^{\ast}{\mu}_{n}$ is also a mean on ${l}^{\mathrm{\infty}}$ for all $n,p\in \mathbb{N}$, and hence $\{{z}_{n,p}\}$ is well defined. By Theorem 3.2, $\{{P}_{F(T)}{x}_{n}\}$ converges strongly to some $w\in F(T)$.
for all $i\in \mathbb{N}$.
for all $i\in \mathbb{N}$. Thus, it follows from the strong regularity of $\{{\mu}_{n}\}$ and (3.19) that ${lim}_{i}{[{\eta}_{i}]}_{k}(f(k+1)f(k))=0$. Hence, $\{{\eta}_{i}\}$ is asymptotically invariant.
By the definitions of $\{{z}_{n,p}\}$ and $\{{\eta}_{i}\}$, we have ${z}_{{n}_{i},{p}_{i}}={G}_{{\eta}_{i}}(\{{x}_{k}\})$ for all $i\in \mathbb{N}$. By Theorem 3.4, $\{{z}_{{n}_{i},{p}_{i}}\}$ converges weakly to w as $i\to \mathrm{\infty}$. This contradicts (3.18). □
As a direct consequence of Theorems 3.4 and 3.5, we obtain the following corollary for a single hybrid mapping.
 (i)
if $\{{\mu}_{n}\}$ is asymptotically invariant, then the sequence ${\{{G}_{{\mu}_{n}}(\{{S}^{k}x\})\}}_{n\in \mathbb{N}}$ converges weakly to the strong limit of $\{{P}_{F(S)}{S}^{n}x\}$;
 (ii)
if $\{{\mu}_{n}\}$ is strongly regular, then the sequence ${\{{G}_{{\mu}_{n}}({\{{S}^{k+p}x\}}_{k})\}}_{n,p\in \mathbb{N}}$ converges weakly to the strong limit of $\{{P}_{F(S)}{S}^{n}x\}$ as $n\to \mathrm{\infty}$ uniformly in $p\in \mathbb{N}$.
4 Consequences of Theorem 3.5
In this section, using the techniques in [2, 4, 6–8], we obtain some consequences of Theorem 3.5. Throughout this section, we suppose that C, H, $({\lambda}_{n})$, λ, $\{{T}_{n}\}$, T, and $\{{x}_{n}\}$ are the same as in Section 3 and ${\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})=F(T)\ne \mathrm{\varnothing}$.
We first obtain the following theorem for Cesàro means of sequences.
Theorem 4.1 The sequence ${\{{(n+1)}^{1}{\sum}_{k=0}^{n}{x}_{k+p}\}}_{n,p\in \mathbb{N}}$ converges weakly to the strong limit of $\{{P}_{F(T)}{x}_{n}\}$ as $n\to \mathrm{\infty}$ uniformly in $p\in \mathbb{N}$.
for each $n,p\in \mathbb{N}$; see, for instance, [2], Theorem 5.1, [4], Theorem 5] and [[8], Section 3.5]. Therefore, Theorem 3.5 implies the conclusion. □
Remark 4.1 In [[10], Theorem 4.1], it was shown that $\{{z}_{n,0}\}$ in Theorem 4.1 converges weakly to the strong limit of $\{{P}_{F(T)}{x}_{n}\}$.
We next obtain the following theorem.
Theorem 4.2 Let $({\rho}_{n})$ be a sequence of $(0,1)$ such that ${\rho}_{n}\to 1$. Then the sequence ${\{(1{\rho}_{n}){\sum}_{k=0}^{\mathrm{\infty}}{\rho}_{n}^{k}{x}_{k+p}\}}_{n,p\in \mathbb{N}}$ converges weakly to the strong limit of $\{{P}_{F(T)}{x}_{n}\}$ as $n\to \mathrm{\infty}$ uniformly in $p\in \mathbb{N}$.
for each $n,p\in \mathbb{N}$; see, for instance, [[2], Theorem 5.2] and [[8], Section 3.5]. Therefore, Theorem 3.5 implies the conclusion. □
By using a strongly regular matrix introduced in [16], we can obtain the following theorem which actually generalizes Theorems 4.1 and 4.2.
Theorem 4.3 Let ${({q}_{n,k})}_{n,k\in \mathbb{N}}$ be a sequence of real numbers such that
(A1) ${q}_{n,k}\ge 0$ for all $n,k\in \mathbb{N}$;
(A2) ${\sum}_{k=0}^{\mathrm{\infty}}{q}_{n,k}=1$ for all $n\in \mathbb{N}$;
(A3) ${lim}_{n}{\sum}_{k=0}^{\mathrm{\infty}}{q}_{n,k}{q}_{n,k+1}=0$.
Then the sequence ${\{{\sum}_{k=0}^{\mathrm{\infty}}{q}_{n,k}{x}_{k+p}\}}_{n,p\in \mathbb{N}}$ converges weakly to the strong limit of $\{{P}_{F(T)}{x}_{n}\}$ as $n\to \mathrm{\infty}$ uniformly in $p\in \mathbb{N}$.
for each $n,p\in \mathbb{N}$; see, for instance, [[2], Theorem 5.3] and [[4], Theorem 7].
for all $n\in \mathbb{N}$.
for all $y\in H$. Thus, (4.6) holds. Therefore, Theorem 3.5 implies the conclusion. □
5 Applications
In this final section, we give two applications of Theorem 4.3. We first obtain a corollary for a single λhybrid mapping; see Corollary 5.1. We next study the problem of finding common fixed points of sequences of nonexpansive mappings; see Corollary 5.3.
for $n\ge 1$. The sequence ${({q}_{n,k})}_{n,k\in \mathbb{N}}$ obviously satisfies (A1)(A3) in Theorem 4.3.
for $n\in \mathbb{N}$. Then ${\{{\sum}_{k=0}^{n}{q}_{n,k}{x}_{k+p}\}}_{n,p\in \mathbb{N}}$ converges weakly to the strong limit of $\{{P}_{F(T)}{x}_{n}\}$ as $n\to \mathrm{\infty}$ uniformly in $p\in \mathbb{N}$.
for all $n\in \mathbb{N}$. Then it is clear that ${x}_{n+1}={T}_{n}{x}_{n}$ for all $n\in \mathbb{N}$ and ${T}_{n}x\to Tx$ for all $x\in C$. It is also clear that $F({T}_{n})=F(T)$ for all $n\in \mathbb{N}$ and hence $\mathrm{\varnothing}\ne F(T)={\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})$.
for all $n\in \mathbb{N}$ and $x,y\in C$. Thus, by setting ${\lambda}_{n}=\lambda +(1\lambda ){\alpha}_{n}$ for all $n\in \mathbb{N}$, we know that ${T}_{n}\in {H}_{{\lambda}_{n}}(C)$ for all $n\in \mathbb{N}$. It is clear that ${\lambda}_{n}\to \lambda $.
Since ${q}_{n,k}=0$ for $k\ge n+1$, it also holds that ${\sum}_{k=0}^{n}{q}_{n,k}{x}_{k+p}={\sum}_{k=0}^{\mathrm{\infty}}{q}_{n,k}{x}_{k+p}$ for all $n,p\in \mathbb{N}$. Consequently, Theorem 4.3 implies the conclusion. □
In order to obtain our final result, we need the following theorem, which was originally shown in strictly convex Banach spaces.
Lemma 5.2 ([[17], Lemma 3])
Let C be a nonempty closed convex subset of a Hilbert space H, $\{{T}_{n}\}$ a sequence of nonexpansive mappings of C into H such that ${\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})$ is nonempty, and $({\gamma}_{n})$ a sequence of $(0,1)$ such that ${\sum}_{k=0}^{\mathrm{\infty}}{\gamma}_{k}=1$. Then the mapping $T={\sum}_{k=0}^{\mathrm{\infty}}{\gamma}_{k}{T}_{k}$ is a nonexpansive mapping of C into H such that $F(T)={\bigcap}_{k=0}^{\mathrm{\infty}}F({T}_{k})$.
and hence T is a selfmapping on C.
As in the proof of [[9], Theorem 3.7], we can show the following corollary.
for $n\in \mathbb{N}$. Then ${\{{\sum}_{k=0}^{n}{q}_{n,k}{x}_{k+p}\}}_{n,p\in \mathbb{N}}$ converges weakly to the strong limit of $\{{P}_{F}{x}_{n}\}$ as $n\to \mathrm{\infty}$ uniformly in $p\in \mathbb{N}$.
for all $n\in \mathbb{N}$. It is clear that ${x}_{n+1}={T}_{n}{x}_{n}$ for all $n\in \mathbb{N}$. Since each ${T}_{n}$ is nonexpansive, we know that ${T}_{n}\in {H}_{1}(C)$ for all $n\in \mathbb{N}$.
as $n\to \mathrm{\infty}$. Thus, ${T}_{n}x\to Tx$. Consequently, Theorem 4.3 implies the conclusion. □
Declarations
Authors’ Affiliations
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