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Research | Open | Published:

Uniform mean convergence theorems for hybrid mappings in Hilbert spaces

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.

1 Introduction

Using the notion of asymptotically invariant sequences of means on l , 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, [28] 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 { μ n } of means on l . In particular, by assuming the strong regularity on { μ 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 ; 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 and , respectively. For a Banach space X, the conjugate space of X is denoted by X . We denote the norms of X and X by . For a sequence { x n } of a Banach space X and xX, strong and weak convergence of { x n } to x are denoted by x n x and x n x, respectively. For a sequence { x n } of X and x X , weak convergence of { x n } to x is also denoted by . The inner product of a Hilbert space H is denoted by ,. For a subset A of a Hilbert space H, the closure of the convex hull of A is denoted by co ¯ A.

Let C be a nonempty subset of a Hilbert space H, and let λR. A mapping T:CH is said to be λ-hybrid [11] if

T x T y 2 x y 2 +2(1λ)xTx,yTy
(2.1)

for all x,yC. It is obvious that the following hold: T is 1-hybrid if and only if it is nonexpansive; T is 0-hybrid if and only if it is nonspreading in the sense of [12]; T is 1/2-hybrid if and only if it is a hybrid mapping in the sense of [13]. It is also known that if T is firmly nonexpansive, then it is λ-hybrid for all λ[0,1]; see [[11], Lemma 3.1]. It should be noted that if T:CH is λ-hybrid for some λ>1, then T is the identity mapping on C. Indeed, by setting x=yC in (2.1), we have

02(1λ) x T x 2 .
(2.2)

Since 1λ<0, we obtain Tx=x.

We denote the set of all λ-hybrid mappings of C into H by H λ (C,H). We also denote by H λ (C) the set of all λ-hybrid mappings of C into itself. The set of all fixed points of a mapping T:CH is denoted by F(T). A mapping T:CH is said to be quasi-nonexpansive if F(T) is nonempty and uTxux for all uF(T) and xC. It is well known that F(T) is closed and convex if T:CH is quasi-nonexpansive and C is closed and convex. It is obvious that if T H λ (C,H) for some λR and F(T) is nonempty, then T is quasi-nonexpansive. We denote the identity mapping on C by I or T 0 , where T:CH is a mapping.

Let C be a nonempty closed convex subset of a Hilbert space H. Then for each xH, there exists a unique z x C such that z x x= min y C yx. The metric projection P C of H onto C is defined by P C x= z x for all xH. For xH and zC, the following holds:

z= P C x sup y C yz,xz0.
(2.3)

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 u x n + 1 u x n for all uS and nN. Then { P S x n } converges strongly.

Let l be the Banach space of all bounded real sequences with supremum norm. For μ ( l ) and f=(f(0),f(1),) l , the value μ(f) is also denoted by

[ μ ] k f(k).
(2.4)

A bounded linear functional μ on l is said to be a mean on l if μ=μ(e)=1, where e=(1,1,). It is known that if μ is a mean on l , then μ(f)μ(g) whenever f,g l satisfy f(k)g(k) for all kN. It is also known that the Hahn-Banach theorem ensures that there exists a mean μ on l such that

[ μ ] k f(k+1)= [ μ ] k f(k)
(2.5)

for all f l , where (f(k+1))=(f(1),f(2),); see [[8], Theorem 1.4.3]. Such a mean μ is called a Banach limit. If μ is a Banach limit and f l is convergent, then μ(f)= lim k f(k).

For pN, the bounded linear operator r p of l into itself is defined by ( r p f)(k)=f(k+p) for all f l and kN. The conjugate operator of r p is denoted by r p ; that is, it is the bounded linear operator of ( l ) into itself defined by ( r p μ)(f)=μ( r p f) for all μ ( l ) and f l . A sequence { μ n } of means on l is said to be asymptotically invariant if , that is,

lim n [ μ n ] k ( f ( k + 1 ) f ( k ) ) =0
(2.6)

for all f l . It is also said to be strongly regular if r 1 μ n μ n 0, that is,

lim n sup f 1 | [ μ n ] k ( f ( k + 1 ) f ( k ) ) | =0.
(2.7)

Some examples of strongly regular sequences of means on l are shown in Sections 4 and 5. See [15] on asymptotically invariant nets of means and [2, 48] on the nonlinear ergodic theory for nonexpansive mappings with asymptotically invariant nets of means. The following lemma is well known.

Lemma 2.2 Let { μ n } be an asymptotically invariant sequence of means on l and { μ n α } a subnet of { μ n } such that . Then μ is a Banach limit.

For the sake of completeness, we give the proof.

Proof Since the norm of ( l ) is weakly lower semicontinuous and for each nN, we have . On the other hand, since and μ n (e)=1 for each nN, we obtain μ(e)= lim α μ n α (e)=1. This implies that 1=μ(e)μ. Hence, μ is a mean on l .

Fix f l . Since and { μ n } is asymptotically invariant, we have

[ μ ] k ( f ( k + 1 ) f ( k ) ) = lim α [ μ n α ] k ( f ( k + 1 ) f ( k ) ) =0.
(2.8)

Thus, μ is a Banach limit. □

Let H be a Hilbert space, μ a mean on l , and { x n } a bounded sequence of H. Since the functional y [ μ ] k x k ,y belongs to H , Riesz’s theorem ensures that there corresponds a unique zH such that

[ μ ] k x k ,y=z,y
(2.9)

for all yH; see [[7], Theorem 1] and [[8], Section 3.3]. We denote such a point z by

G ( { x k } , μ ) or G μ ( { x k } ) .
(2.10)

In other words, it is a unique element of H such that

[ μ ] k x k ,y= G ( { x k } , μ ) , y
(2.11)

for all yH. 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 pH, then G({ x k },μ)=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 ( β n ) a convergent sequence of real numbers. Then [ μ ] n ( β n x n x n + 1 , y n )=0 for each Banach limit μ.

Proof Let μ be a Banach limit. Set y= lim n y n and β= lim n β n . Since μ is a Banach limit and the second and third terms of the right-hand side of the equality

(2.12)

tend to 0, we have [ μ ] n ( β n x n x n + 1 , y n )=β [ μ ] n x n x n + 1 ,y=0. □

3 Mean convergence theorems

In this section, we show mean convergence theorems for a pointwise convergent sequence of mappings in λ R H λ (C).

Throughout this section, we suppose the following conditions:

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

  • ( λ n ) is a sequence of real numbers which tends to λR;

  • { T n } is a sequence of mappings such that T n H λ n (C) for all nN and { T n x} converges strongly for all xC;

  • T is a mapping of C into itself defined by Tx= lim n T n x for all xC;

  • { x n } is a sequence of C defined by x 0 C and x n + 1 = T n x n for all nN.

Motivated by [710, 12], we first show the following fundamental theorem.

Theorem 3.1 If { x n } is bounded, then G({ x k },μ) is a fixed point of T for each Banach limit μ.

Proof Let μ be a Banach limit. Set z=G({ x k },μ). Since and C is closed and convex, we have zC. By assumption,

M= sup n N ( T n z T z + 2 x n + 1 T n z )
(3.1)

is finite. Since each T n is λ n -hybrid, we have

(3.2)

for all nN. By Lemma 2.3, we have

[ μ ] n ( ( 1 λ n ) x n x n + 1 , z T n z ) =0.
(3.3)

By (3.2), (3.3), and T n zTz, we obtain

[ μ ] n x n T z 2 = [ μ ] n x n + 1 T z 2 [ μ ] n x n z 2 .
(3.4)

On the other hand, by the definition of z, we also know that

[ μ ] n x n z 2 = [ μ ] n ( x n T z 2 + T z z 2 + 2 x n T z , T z z ) = [ μ ] n x n T z 2 + T z z 2 + 2 z T z , T z z = [ μ ] n x n T z 2 T z z 2 .
(3.5)

It follows from (3.4) and (3.5) that 0 T z z 2 . Therefore, z is a fixed point of T. □

Using Lemma 2.1 and Theorem 3.1, we next show the following theorem.

Theorem 3.2 Suppose that F(T) is nonempty and F(T)= n = 0 F( T n ). Then { x n } is bounded, { P F ( T ) x n } is strongly convergent, and

G ( { x k } , μ ) = lim n P F ( T ) x n
(3.6)

for each Banach limit μ.

Proof Let μ be a Banach limit. It is obvious that T H λ (C). Hence, F(T) is a nonempty closed convex subset of H, and hence P F ( T ) is well defined. We denote P F ( T ) by P. Since each T n is quasi-nonexpansive and F(T)F( T n ), we have

u x n + 1 =u T n x n u x n
(3.7)

for all uF(T) and nN. It also follows from (3.7) that { x n } is bounded. According to Theorem 3.1, we know that G({ x k },μ) is a fixed point of T. Using Lemma 2.1 and (3.7), we also know that {P x n } converges strongly to some wF(T).

Set z=G({ x k },μ). By the definition of P and (3.7), we have

P x n + 1 x n + 1 P x n x n + 1 P x n x n
(3.8)

for all nN. On the other hand, it follows from zF(T) and (2.3) that

zP x n , x n P x n 0
(3.9)

for all nN. This gives us that

(3.10)

for all nN. By (3.8) and (3.10), we have

zw, x n P x n P x n w x 0 P x 0
(3.11)

for all nN. Consequently, we obtain

z w 2 = [ μ ] n z w , x n lim n z w , P x n = [ μ ] n z w , x n [ μ ] n z w , P x n = [ μ ] n z w , x n P x n [ μ ] n ( P x n w x 0 P x 0 ) = 0 .
(3.12)

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.

Corollary 3.3 Suppose that xC and S H γ (C) for some γR. Then the following hold:

  1. (i)

    if { S n x} is bounded, then F(S) is nonempty and G({ S k x},μ) is a fixed point of S for each Banach limit μ;

  2. (ii)

    if F(S) is nonempty, then { S n x} is bounded, { P F ( S ) S n x} is strongly convergent, and

    G ( { S k x } , μ ) = lim n 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 , we next show the following mean convergence theorem.

Theorem 3.4 Suppose that F(T) is nonempty and F(T)= n = 0 F( T n ). Let { μ n } be an asymptotically invariant sequence of means on l . Then the sequence

{ G μ n ( { x k } ) } n N
(3.14)

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 wF(T).

Let { z n } be the sequence defined by z n = G μ n ({ x k }) for all nN. Since for all nN, 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 u. It follows from μ n i =1 that there exists a subnet { μ n i α } of { μ n i } such that . Since { μ n i } is asymptotically invariant, Lemma 2.2 implies that μ is a Banach limit.

By Theorem 3.2, we know that

G ( { x k } , μ ) = lim n P F ( T ) x n =w.
(3.15)

This gives us that

z n i α ,y= [ μ n i α ] k x k ,y [ μ ] k x k ,y= G ( { x k } , μ ) , y =w,y
(3.16)

for all yH. Thus, { z n i α } converges weakly to w. On the other hand, since z n i u and { z n i α } is a subnet of { z n i }, we know that z n i α 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 { μ n } is assumed.

Theorem 3.5 Suppose that F(T) is nonempty and F(T)= n = 0 F( T n ). Let { μ n } be a strongly regular sequence of means on l . Then the sequence

{ G r p μ n ( { x k } ) } n , p N
(3.17)

converges weakly to the strong limit of { P F ( T ) x n } as n uniformly in pN.

Proof Set z n , p = G r p μ n ({ x k }) for all n,pN. It is easy to see that r p μ n is also a mean on l for all n,pN, and hence { z n , p } is well defined. By Theorem 3.2, { P F ( T ) x n } converges strongly to some wF(T).

We show that for each yH and ε>0, there exists NN such that n,pN and nN imply that | z n , p w,y|<ε. Suppose that this assertion does not hold. Then there exist y 0 H, ε 0 >0, a strictly increasing sequence { n i } of , and a sequence { p i } of such that

| z n i , p i w , y 0 | ε 0
(3.18)

for all iN.

Set η i = r p i μ n i for all iN. Then { η i } is asymptotically invariant. Indeed, if f l , then we have

(3.19)

for all iN. Thus, it follows from the strong regularity of { μ n } and (3.19) that lim i [ η i ] k (f(k+1)f(k))=0. Hence, { η i } is asymptotically invariant.

By the definitions of { z n , p } and { η i }, we have z n i , p i = G η i ({ x k }) for all iN. By Theorem 3.4, { z n i , p i } converges weakly to w as i. 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.

Corollary 3.6 Suppose that xC, S H γ (C) for some γR, and F(S) is nonempty. Let { μ n } be a sequence of means on l . Then the following hold:

  1. (i)

    if { μ n } is asymptotically invariant, then the sequence { G μ n ( { S k x } ) } n N converges weakly to the strong limit of { P F ( S ) S n x};

  2. (ii)

    if { μ n } is strongly regular, then the sequence { G μ n ( { S k + p x } k ) } n , p N converges weakly to the strong limit of { P F ( S ) S n x} as n uniformly in pN.

4 Consequences of Theorem 3.5

In this section, using the techniques in [2, 4, 68], we obtain some consequences of Theorem 3.5. Throughout this section, we suppose that C, H, ( λ n ), λ, { T n }, T, and { x n } are the same as in Section 3 and n = 0 F( T n )=F(T).

We first obtain the following theorem for Cesàro means of sequences.

Theorem 4.1 The sequence { ( n + 1 ) 1 k = 0 n x k + p } n , p N converges weakly to the strong limit of { P F ( T ) x n } as n uniformly in pN.

Proof Let { μ n } be the sequence of means on l defined by

μ n (f)= 1 n + 1 k = 0 n f(k)
(4.1)

for all nN and f l . It is well known that { μ n } is strongly regular and

G r p μ n ( { x k } ) = 1 n + 1 k = 0 n x k + p
(4.2)

for each n,pN; 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 ( ρ n ) be a sequence of (0,1) such that ρ n 1. Then the sequence { ( 1 ρ n ) k = 0 ρ n k x k + p } n , p N converges weakly to the strong limit of { P F ( T ) x n } as n uniformly in pN.

Proof Let { μ n } be the sequence of means on l defined by

μ n (f)=(1 ρ n ) k = 0 ρ n k f(k)
(4.3)

for all nN and f l . It is well known that { μ n } is strongly regular and

G r p μ n ( { x k } ) =(1 ρ n ) k = 0 ρ n k x k + p
(4.4)

for each n,pN; 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 N be a sequence of real numbers such that

(A1) q n , k 0 for all n,kN;

(A2) k = 0 q n , k =1 for all nN;

(A3) lim n k = 0 | q n , k q n , k + 1 |=0.

Then the sequence { k = 0 q n , k x k + p } n , p N converges weakly to the strong limit of { P F ( T ) x n } as n uniformly in pN.

Proof Let { μ n } be the sequence of means on l defined by

μ n (f)= k = 0 q n , k f(k)
(4.5)

for all nN and f l . It is well known that { μ n } is strongly regular and

G r p μ n ( { x k } ) = k = 0 q n , k x k + p
(4.6)

for each n,pN; see, for instance, [[2], Theorem 5.3] and [[4], Theorem 7].

For the sake of completeness, we give the proof of this fact. It follows from (A1) that

q n , 0 =| q n , 0 | k = 0 m | q n , k q n , k + 1 |+ q n , m + 1
(4.7)

for all n,mN. It follows from (A2) that lim k q n , k =0 for all nN. Thus, letting m in (4.7), we have

q n , 0 k = 0 | q n , k q n , k + 1 |
(4.8)

for all nN. It also holds that

(4.9)

for all nN.

By (4.8), (4.9), and (A3), we have

(4.10)

and hence { μ n } is strongly regular. On the other hand, if n,pN, then we have

[ r p μ n ] k x k ,y= [ μ n ] k x k + p ,y= k = 0 q n , k x k + p ,y= k = 0 q n , k x k + p , y
(4.11)

for all yH. 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 λ-hybrid mapping; see Corollary 5.1. We next study the problem of finding common fixed points of sequences of nonexpansive mappings; see Corollary 5.3.

Throughout this section, we suppose that ( β n ) is a sequence of (0,1) satisfying β n 0 and ( q n , k ) n , k N is the sequence of real numbers defined by q 0 , 0 =1, q 0 , k =0 (k1), and

q n , k ={ n 1 ( 1 β n ) ( 0 k n 1 ) ; β n ( k = n ) ; 0 ( k n + 1 )

for n1. The sequence ( q n , k ) n , k N obviously satisfies (A1)-(A3) in Theorem 4.3.

Corollary 5.1 Let C be a nonempty closed convex subset of a Hilbert space H, T H λ (C) for some λR such that F(T) is nonempty, and ( α n ) a sequence of [0,1) such that α n 0. Let { x n } be the sequence of C defined by x 0 C and

x n + 1 = α n x n +(1 α n )T x n

for nN. Then { k = 0 n q n , k x k + p } n , p N converges weakly to the strong limit of { P F ( T ) x n } as n uniformly in pN.

Proof Let { T n } be the sequence of mapping of C into itself defined by

T n = α n I+(1 α n )T
(5.1)

for all nN. Then it is clear that x n + 1 = T n x n for all nN and T n xTx for all xC. It is also clear that F( T n )=F(T) for all nN and hence F(T)= n = 0 F( T n ).

Since T H λ (C), we know that

for all nN and x,yC. Thus, by setting λ n =λ+(1λ) α n for all nN, we know that T n H λ n (C) for all nN. It is clear that λ n λ.

Since q n , k =0 for kn+1, it also holds that k = 0 n q n , k x k + p = k = 0 q n , k x k + p for all n,pN. 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 n = 0 F( T n ) is nonempty, and ( γ n ) a sequence of (0,1) such that k = 0 γ k =1. Then the mapping T= k = 0 γ k T k is a nonexpansive mapping of C into H such that F(T)= k = 0 F( T k ).

Remark 5.1 If T n (C)C for all nN in Lemma 5.2, then T(C)C. Indeed, for each xC, we have

Tx= lim N 1 j = 0 N γ j k = 0 N γ k T k xC
(5.2)

and hence T is a self-mapping on C.

As in the proof of [[9], Theorem 3.7], we can show the following corollary.

Corollary 5.3 Let C be a nonempty closed convex subset of a Hilbert space H, { S n } a sequence of nonexpansive mappings of C into itself such that F= k = 0 F( S k ) is nonempty, and ( γ n ) a sequence of (0,1) such that n = 0 γ n =1. Let { x n } be the sequence of C defined by x 0 C and

x n + 1 = k = 0 n γ k S k x n + ( 1 k = 0 n γ k ) S n + 1 x n
(5.3)

for nN. Then { k = 0 n q n , k x k + p } n , p N converges weakly to the strong limit of { P F x n } as n uniformly in pN.

Proof Let { T n } be the sequence of mappings of C into itself defined by

T n = k = 0 n γ k S k + ( 1 k = 0 n γ k ) S n + 1
(5.4)

for all nN. It is clear that x n + 1 = T n x n for all nN. Since each T n is nonexpansive, we know that T n H 1 (C) for all nN.

By Lemma 5.2 and Remark 5.1, the mapping T= k = 0 γ k S k is a nonexpansive mapping of C into itself such that F(T)=F. Since F is nonempty by assumption, so is F(T). By Lemma 5.2, we also know that F( T n )= k = 0 n + 1 F( S k ) and hence we have

n = 0 F( T n )= n = 0 k = 0 n + 1 F( S k )=F=F(T).
(5.5)

It remains to be seen that T n xTx for all xC. Fix xC. Since F is nonempty, we can fix pF. Since for all kN, we know that is finite. By k = 0 γ k =1 and the definitions of T and T n , we also know that

(5.6)

as n. Thus, T n xTx. Consequently, Theorem 4.3 implies the conclusion. □

References

  1. 1.

    Baillon J-B: Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert. C. R. Acad. Sci. Paris Sér. A-B 1975, 280: 1511–1514.

  2. 2.

    Atsushiba S, Lau AT, Takahashi W: Nonlinear strong ergodic theorems for commutative nonexpansive semigroups on strictly convex Banach spaces. J. Nonlinear Convex Anal. 2000, 1: 213–231.

  3. 3.

    Brézis H, Browder FE: Nonlinear ergodic theorems. Bull. Am. Math. Soc. 1976, 82: 959–961. 10.1090/S0002-9904-1976-14233-4

  4. 4.

    Hirano N, Kido K, Takahashi W: Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces. Nonlinear Anal. 1988, 12: 1269–1281. 10.1016/0362-546X(88)90059-4

  5. 5.

    Lau AT, Shioji N, Takahashi W: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces. J. Funct. Anal. 1999, 161: 62–75. 10.1006/jfan.1998.3352

  6. 6.

    Rodé G: An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space. J. Math. Anal. Appl. 1982, 85: 172–178. 10.1016/0022-247X(82)90032-4

  7. 7.

    Takahashi W: A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space. Proc. Am. Math. Soc. 1981, 81: 253–256. 10.1090/S0002-9939-1981-0593468-X

  8. 8.

    Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.

  9. 9.

    Akatsuka M, Aoyama K, Takahashi W: Mean ergodic theorems for a sequence of nonexpansive mappings in Hilbert spaces. Sci. Math. Jpn. 2008, 68: 233–239.

  10. 10.

    Aoyama K, Kohsaka F: Fixed point and mean convergence theorems for a family of λ -hybrid mappings. J. Nonlinear Anal. Optim. 2011, 2: 85–92.

  11. 11.

    Aoyama K, Iemoto S, Kohsaka F, Takahashi W: Fixed point and ergodic theorems for λ -hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2010, 11: 335–343.

  12. 12.

    Kohsaka F, Takahashi W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. 2008, 91: 166–177. 10.1007/s00013-008-2545-8

  13. 13.

    Takahashi W: Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 2010, 11: 79–88.

  14. 14.

    Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417–428. 10.1023/A:1025407607560

  15. 15.

    Day MM: Amenable semigroups. Ill. J. Math. 1957, 1: 509–544.

  16. 16.

    Lorentz GG: A contribution to the theory of divergent sequences. Acta Math. 1948, 80: 167–190. 10.1007/BF02393648

  17. 17.

    Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 1973, 179: 251–262.

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Correspondence to Fumiaki Kohsaka.

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The authors declare that they have no competing interests.

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Keywords

  • ergodic theorem
  • hybrid mapping
  • nonexpansive mapping
  • nonspreading mapping
  • uniform mean convergence theorem