Application of the product net technique and Kadec–Klee property to study nonlinear ergodic theorems and weak convergence theorems in uniformly convex multi-Banach spaces

R E S E A R C H Open Access

Application of the product net technique and Kadec–Klee property to study nonlinear ergodic theorems and weak convergence theorems in uniformly convex multi-Banach spaces

H.M. Kenari¹, Reza Saadati^2* and Choonkil Park³

*Correspondence:rsaadati@eml.cc

2Department of Mathematics, Iran University of Science and Technology, Tehran, Iran Full list of author information is available at the end of the article

Abstract

LetYbe a uniformly convex multi-Banach space which has not a Frechet

differentiable norm. We use the technique of product net to obtain the nonlinear ergodic theorems inY. Finally, let the dual of uniformly convex multi-Banach space have the Kadec–Klee property, we instate the weak convergence theorem in the case of reversible semi-group.

MSC: Primary 39A10; 39B72; secondary 47H10; 46B03

Keywords: Reversible semi-groups; Kadec–Klee property; Asymptotically nonexpansive mapping; Almost orbit; Uniformly convex multi-Banach space

1 Preliminaries

Dales and Polyakov in [1] introduced a multi–normed space by using the concept of opera- tor sequence space, operator spaces, and Banach lattices; for more details and application, we refer to [1–3].

In this paper assume that (Y, · ) is a complex normed space, and let∈N. We denote byYthe vector spaceY⊕ · · · ⊕Yconsisting of-tuples (y1, . . . ,y), wherey1, . . . ,y∈Y. The linear operations onYare defined coordinate-wise. The zero element of eitherYor Yis denoted by 0. We denote byNthe set{1, 2, . . . ,}and byΣthe group of permuta- tions onsymbols.

Definition 1.1 Suppose thatY is a vector space, and take∈N. Forσ∈Σ, define Bσ(y) = (yσ(1), . . . ,y_σ()), y= (y₁, . . . ,y)∈Y.

Forβ= (β_j)∈C, define

Kβ(y) = (β_jy_j), y= (y₁, . . . ,y)∈Y.

©The Author(s) 2019. 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, pro- vided 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.

Definition 1.2 Assume that (Y, · ) is a complex (respectively, real) normed space, and takem∈N. A multi-norm of levelmon{Y:∈Nm}is a sequence ( · :∈Nm) such that · is a norm onYfor each∈Nm, such thaty1=yfor eachy∈Y (so that · 1is the initial norm), and such that the following Axioms (a1)–(a4) are satisfied for each∈Nmwithk≥2:

(a1) for eachσ∈Σandy∈Y, we have Bσ(y)

=y;

(a2) for eachβ₁, . . . ,β∈C(respectively, eachβ₁, . . . ,β∈R) andy∈Y, we have Kβ(y)

≤ maxj∈N

|β_j| y; (a3) for eachy₁, . . . ,y_–1, we have

(y1, . . . ,y–1, 0)

=(y1, . . . ,y–1)

–1; (a4) for eachy1, . . . ,y–1∈Y,

(y₁, . . . ,y_–2,y_–1,y_–1)

=(y₁, . . . ,y_–1)

–1.

In this case, ((Y, · ) :∈Nm) is a multi-normed space of levelm.

A multi-norm on{Y:∈N}is a sequence ·

=

· :∈N

such that ( · :∈ Nm) is a multi-norm of level m for each m∈N. In this case, ((Y^m, · m) :m∈N) is a multi-normed space.

Lemma 1.3([3]) Let((Y, · ) :∈N)be a multi-normed space,and take∈Nm.Then (a) (y, . . . ,y)=y(y∈Y);

(b) max_j∈Ny_j ≤ (y₁, . . . ,y)≤

j=1y_j ≤max_j∈Ny_j(y₁, . . . ,y∈Y).

It follows from (b) that if (Y, · ) is a Banach space, then (Y, · ) is a Banach space for each∈N; in this case ((Y, · ) :∈N) is a multi-Banach space.

Example1.4 ([1]) The sequence ( · :∈N) on{Y:∈N}defined by (y1, . . . ,y)

:=max

j∈N

yj (y1, . . . ,y∈Y) is a multi-norm called the minimum multi-norm.

Example1.5 ([1]) Assume that{( · ^β :∈N) :β ∈B} is the (non-empty) family of all multi-norms on{Y:∈N}. For∈N, set

(y1, . . . ,y)

k:=sup

β∈B

(y1, . . .y)^β

(y1, . . . ,y∈Y).

Then ( · :∈N) is a multi-norm on{Y:∈N}, called the maximum multi-norm.

By the property (b) of multi-norms and the triangle inequality for the norm·k, we can get the following properties. Suppose that ((Y, · ) :∈N) is a multi-normed space. Let ∈Nand (y1, . . . ,y)∈Y^k. For everyi∈ {1, . . . ,}, let (yⁱ_m)m=1,2,...be a sequence inY such thatlim_m→∞yⁱ_m=yi. Then for each (z₁, . . . ,z)∈Ywe have

m→∞lim

y¹_m–z₁, . . . ,y_m–z

= (y₁–z₁, . . . ,y–z).

A sequence (y_m) inYis amulti-nullsequence if, for everyε> 0, there existsm0∈Nsuch that

sup

∈N

(yn, . . . ,ym+–1)

<ε (m≥m0).

Lety∈Y. We say that the sequence (ym) ismulti-convergenttoy∈Yand write

m→∞lim y_m=y

when (y_m–y) is a multi-null sequence.

Assume that G is a semi-topological semi-group. In this article, C is a nonempty bounded closed convex subset of a uniformly convex Banach spaceX. LetX^∗be the dual ofX, then the value ofu^∗∈X^∗atu∈Xwill be denoted byu,u^∗, and we associate the set

J(u) =

u^∗∈X: u,u^∗

=u²=u^∗2 .

It is clear from the Hahn–Banach theorem thatJ(u) is not empty for allu∈X. Then the multi-valued operatorJ:X→X^∗is called the normalized duality mapping ofX, alsok= {Jk(t) :t∈G}is a reversible semigroup of asymptotically nonexpansive functions acting onC. LetF(k) denote the set of all fixed points ofk, i.e.,F(k) ={u∈C:J_k(t)u=u,∀t∈ G}. For each> 0 andp∈G, we put

F

Jk(p)

=

u∈C:J1(p)u–u, . . . ,Jk(p)u–u

k≤

.

Note that if, for any> 0, there existsp∈Gsuch that for allp>p,u∈F(Jk(p)), then lim_p∈GJk(p)u=u; moreover,u∈F(k) by the continuity of elements {Jk(p),p∈G}(for more details, we refer to [4–9]).

We denote the set of all almost orbits ofkand the set{J_k(p)u_k(·) :p∈G,u_k∈AO(k)} byAO(k) andLAO(k), respectively. Denote byωω(uk) the set of all weak limit points of subnets of net{u_k(t)}t∈G.

Lemma 1.6([10]) Assume that X is a Banach space and J is the normalized duality func- tion.Therefore

u+v²≤ u²+ 2v,j(u+v) for all j(u+v)∈J(u+v)and u,v∈X.

Lemma 1.7([11]) Assume that{(X^k, · k)}k∈Nis a uniformly convex multi-Banach space and∅ =C⊂X^kis a bounded closed convex set.Then there exists a strictly increasing con- tinuous convex functionξ: [0, +∞)→[0, +∞)withξ(0) = 0such that

ξ

J1

_n

i=1

aiui

–

n i=1

aiJ1ui, . . . ,Jk

_n

i=1

aiui

–

n i=1

aiJkui

k

≤ max

1≤i,j≤n

ui–uj–(J1ui–J1uj, . . . ,Jkui–Jkuj)

k

for all integers a1, . . . ,an≥0,n≥1withn

i=1ai= 1,u1, . . . ,un∈C,and every nonexpansive function Jkof C to C.

Lemma1.7implies that, for alla1, . . . ,an≥0 withn

i=1ai= 1,u1, . . . ,un∈C,

J1(p) _n

i=1

aiui

–

n i=1

aiJ1(p)u_i, . . . ,Jk(p) _n

i=1

aiui

–

n i=1

aiJk(p)u_i

k

≤

1 +α(p) ξ^–1

1≤i,j≤nmax

u_i–u_j

– 1

1 +α(p)J1(p)ui–J1(p)uj, . . . ,Jk(p)ui–Jk(p)uj

k

≤

1 +α(p) ξ^–1

1≤i,j≤nmax

u_i–u_j

–J1(p)u_i–J1(p)u_j, . . . ,J_k(p)u_i–J_k(p)u_j

k

+d·α(p)

in whichd= 4sup{u:u∈C}+ 1.

For every∈(0, 1], define

a() =min ²

(d+ 2)², ³ (3d+ 2)²ξ

4

and G=

h∈G:α(p)≤ ,

in whichξ(·) is as Lemma1.7. ThenG=∅for> 0, and ifp∈G, then for allt≥p,t∈G. Note thatGa()⊂Gfor all∈(0, 1].

2 Main result

For studies on ergodic theory and its history, we refer to [4–30]. The results of this paper are an extension and generalization of [31].

Lemma 2.1 For all p∈Ga(), coFa()

Jk(p)

⊂F

JK(p) .

Proof SinceF(JK(p)) is closed, we only need to prove that, for allp∈Ga(), coFa()

Jk(p)

⊂F

JK(p) . Letv=n

i=1a_iv_i,v_i∈F_a()(J_k(p)),a_i≥0,i= 1, . . . ,n, andn

i=1a_i= 1. Then J1(p)v–v, . . . ,Jk(p)v–v

k

=

J1(p) n

i=1

aivi– n

i=1

aivi, . . . ,Jk(p) n

i=1

aivi– n

i=1

aivi

k

≤

J₁(p) n

i=1

a_iv_i– n

i=1

a_iJ₁(p)v_i, . . . ,J_k(p) n

i=1

a_iv_i– n

i=1

a_iJ_k(p)v_i

k

≤2ξ^–1

1≤i,j≤nmax

vi–vj–J1(p)vi–J1(p)vi, . . . ,Jk(p)vi–Jk(p)vi

k

+d·α(p)

+a()

≤2ξ^–1

1≤i,j≤nmaxvi–J1(p)vi, . . . ,vi–Jk(p)vi

k+vj–J1(p)vj, . . . ,vj–Jk(p)vj

k

+d·α(p)

+a()

≤2ξ^–1

2a() +d·a() +a()

≤ 2+

2=.

Lemma 2.2 For every p∈G₄,

F₄ Jk(p)

+B

0, 4

⊂F

Jk(p) .

Proof Letp∈G₄ andu=v+w∈F₄(Jk(p)) +B(0,₄), wherev∈F₄(Jk(p)) andw∈B(0,₄), then

J₁(p)u–u, . . . ,J_k(p)u–u

k

=J1(p)(v+w) – (v+w), . . . ,Jk(p)(v+w) – (v+w)

k

≤J1(p)(v+w) –J1(p)v, . . . ,Jk(p)(v+w) –Jk(p)v

k

+J1(p)v–v, . . . ,Jk(p)v–v

k+w

≤2w+J1(p)v–v, . . . ,Jk(p)v–v

k+w

≤3 4+

4 =.

Lemma 2.3 Assume that∈(0, 1]and p∈Ga(a(₄)),so we can find n0∈N such that,for each n≥n0and u∈C,

1 n

n i=1

Jk

pⁱ u∈F

Jk(p) .

Proof Let∈(0, 1] andm=^2d+1_a(

4). There isn0∈Nsatisfying n0≥max

12md

, 32m²d(d+ 1)

ξ a(₄)

2

–1

. For anyn≥n0andp∈G_a(a(4)), we can take a number

K=m²d

1 + 2nα(p) ξ

a(₄) 2

–1 k<n

2

. For everyi∈Nandu∈C, we put

ai(u) =ξ 8

9

1 m

m j=1

J1

p^i+j+1

u–J1(p)1 m

m j=1

J1

p^i+j u, . . . ,

1 m

m j=1

J_k p^i+j+1

u–J_k(p)1 m

m j=1

J_k p^i+j

u

k

.

Byα(p)≤¹₈and a_i(u)≤ max

1≤j,t≤mJ₁ p^i+j

u–J_k p^i+t

u, . . . ,J_k p^i+j

u–J_k p^i+t

u

k

–J₁ p^i+j+1

u–J_k p^i+t+1

u, . . . ,J_k p^i+j+1

u–J_k p^i+t+1

u

k+d·α(p)

≤

1≤j<t≤m

J1

p^i+j u–J1

p^i+t u, . . .Jk

p^i+j u–Jk

p^i+t u

k

–J1

p^i+j+1 u–J1

p^i+t+1 u, . . .Jk

p^i+j+1 u–Jk

p^i+t+1 u

k+dα(p) , we get

n i=1

ai(u)

≤ n

i=1

1≤j<t≤m

J1

p^i+j u–Jk

p^i+t u, . . . ,Jk

p^i+j u–Jk

p^i+t u

k

–J1

p^i+j+1 u–Jk

p^i+t+1 u, . . . ,Jk

p^i+j+1 u–Jk

p^i+t+1 u

k+d·α(p)

=

1≤j<t≤m

n i=1

J1

p^i+j u–Jk

p^i+t u, . . . ,Jk

p^i+j u–Jk

p^i+t u

k

–J1

p^i+j+1 u–Jk

p^i+t+1 u, . . . ,Jk

p^i+j+1 u–Jk

p^i+t+1 u

k+d·α(p)

≤

1≤j<t≤m

d+nd·α(p)

≤m²d

1 +nα(p) .

Suppose that there is an element saytin{ai(u) :i= 1, 2, . . . , 2n}such that ifai(u)≥ξ(^a(₂⁴⁾), then

tξ a(₄)

2

≤m²d

1 + 2nα(p) .

Hence

t≤m²d

1 + 2nα(p) ξ

a(₄) 2

–1

=K.

So, there are at mostN= [K] terms in{a_i(u) :i= 1, 2, . . . , 2n}witha_i(u)≥ξ(^a(₂⁴⁾). Then, for everyiin{1, 2, . . . ,n}, there exists at least one termai+j0(u) (0≤j0≤N) in{ai+j(u) :j= 0, 1, . . . ,N}holdai+j0<ξ(^a(₂⁴⁾).

Put

_i=min

j:a_i+j(u) <ξ a(₄)

2

, 0≤j≤N

,

i= 1, 2, . . . ,n. Now, there are at mostNelements in{i= 1, 2, . . . ,n}such thati= 0. Since

J₁(p)1 m

m j=1

J₁ p^i+i+j

u– 1 m

m j=1

J₁ p^i+i+j

u, . . . ,

Jk(p)1 m

m j=1

Jk

p^i+i+j u– 1

m m

j=1

Jk

p^i+i+j u

k

≤

J₁(p)1 m

m j=1

J₁ p^i+i+j

u– 1 m

m j=1

J₁

p^i+i+j+1 u, . . . ,

Jk(p)1 m

m j=1

Jk

p^i+i+j u– 1

m m

j=1

Jk

h^i+i+j+1 u

k

+

1 m

m j=1

J1

p^i+i+j u– 1

m m

j=1

J1

p^i+i+j+1 u, . . . ,

1 m

m j=1

J_k p^i+i+j

u– 1 m

m j=1

J_k

p^i+i+j+1 u

k

≤9 8ξ^–1

ai+i(u) + d

2m

≤ 9 16a

4

+1

4a

4

<a

4

,

we can conclude that, for allp∈G_a(a(4)), 1

m m

j=1

Jk

p^i+i+j

u∈Fa(₄)

Jk(p) .

By Lemma2.1, we get, for allp∈G_a(a(4))⊂G_a(4), 1

n n

i=1

1 m

m j=1

Jk

p^i+i+j

u∈coFa(₄)

Jk(p)

⊂F₄ Jk(p)

.

Using Lemma2.2and

1 n

n i=1

J1

pⁱ u–1

n n

i=1

1 m

m j=1

J1

p^i+i+j u, . . . ,

1 n

n i=1

J_k pⁱ

u–1 n

n i=1

1 m

m j=1

J_k p^i+i+j

u

k

≤ 1 mn

m j=1

_n

i=1

J1

pⁱ u–

n i=1

J1

p^i+i+j u, . . . ,

n i=1

Jk

pⁱ

u–sumⁿ_i=1Jk

p^i+i+j u

k

≤ 1 mn

m j=1

_n

i=1

J1

pⁱ u–

n i=1

J1

p^i+j u, . . . ,

n i=1

Jk

pⁱ u–

n i=1

Jk

p^i+j u

k

+ 1 mn

m j=1

_n

i=1

L₁ p^i+j

u– n

i=1

J₁ p^i+i+j

u, . . . , n

i=1

Jk

p^i+j u–

n i=1

Jk

p^i+i+j u

k

≤md n +Nd

n

≤

12+m²d²(ξ(^a(

4) 2 ))^–1

n + 2m²d²α(p)

ξ a(₄)

2 –1

<

12+ 32+

8<

4, we obtain

1 n

n i=1

J_k pⁱ

u∈F

4

J_k(p) +B

0,

4

⊂F

J_k(p)

.

Lemma 2.4 Suppose that uk(·)is an almost orbit ofk.So limt∈Gγu₁(t) + (1 –γ)ϕ–g, . . . ,γu_k(t) + (1 –γ)ϕ–g

k

exist for everyγ ∈(0, 1)andϕ,g∈F(k).

Proof To complete the proof, it is enough to prove that

s∈Ginfsup

t∈G

γu₁(ts) + (1 –γ)ϕ–g, . . . ,γu_k(ts) + (1 –γ)ϕ–g

k

≤sup

s∈Ginf

t∈Gγu1(ts) + (1 –γ)ϕ–g, . . . ,γuk(ts) + (1 –γ)ϕ–g

k.

We know, for every> 0, there aret0ands0∈Gsuch that, for anyt∈G,α(tt0) <_1+d and ϕ(ts0) <, whereϕ(t) =sup_p∈G(u1(pt) –J1(p)u1(t), . . . ,uk(pt) –Jk(p)uk(t))k. So, for every

a∈G,

s∈Ginfsup

t∈G

u₁(tss₀) –ϕ, . . . ,u_k(tss₀) –ϕ

k

≤sup

t∈G

u1(tt0as0) –ϕ, . . . ,uk(tt0as0) –ϕ

k

≤sup

t∈G

u1(tt0as0) –J1(tt0)u1(as0), . . . ,uk(tt0as0) –Jk(tt0)uk(as0)

k

+sup

t∈G

J1(tt0)u1(as0) –ϕ, . . . ,Jk(tt0)uk(as0) –ϕ

k

≤ϕ(as₀) +sup

t∈G

1 +α(tt₀)

·u₁(as₀) –ϕ, . . . ,u_k(as₀) –ϕ

k

≤u1(as0) –ϕ, . . . ,uk(as0) –ϕ

k+ 2.

Hence

s∈Ginfsup

t∈G

u1(tss0) –ϕ, . . . ,uk(tss0) –ϕ

k≤inf

a∈Gu1(as0) –ϕ, . . . ,uk(as0) –ϕ

k+ 2.

Thus, there existss1∈Gsuch that sup

t∈G

u₁(ts₁s₀) –ϕ, . . . ,u_k(ts₁s₀) –ϕ

k<inf

a∈Gu₁(as₀) –ϕ, . . . ,u_k(as₀) –ϕ

k+ 3.

Then, for everya∈G, we get

s∈Ginfsup

t∈G

γu1(ts) + (1 –γ)ϕ–g, . . . ,γuk(ts) + (1 –γ)ϕ–g

k

≤sup

t∈G

γu₁(tt₀as₁s₀) + (1 –γ)ϕ–g, . . . ,γu_k(tt₀as₁s₀) + (1 –γ)ϕ–g

k

≤γsup

t∈G

u1(tt0as1s0) –J1(tt0)u1(as1s0), . . . ,uk(tt0as1s0) –Jk(tt0)uk(as1s0)

k

+sup

t∈G

γJ1(tt0)u1(as1s0) + (1 –γ)ϕ–g, . . . ,γJk(tt0)uk(as1s0) + (1 –γ)ϕ–g

k

≤ϕ(as₁s₀) +sup

t∈G

γJ₁(tt₀)u₁(as₁s₀) + (1 –γ)ϕ–J₁(tt₀)

γu₁(as₁s₀) + (1 –γ)ϕ , . . . , γJk(tt0)uk(as1s0) + (1 –γ)ϕ–Jk(tt0)

γuk(as1s0) + (1 –γ)ϕ

k

+sup

t∈G

J₁(tt₀)

γu₁(as₁s₀) + (1 –γ)ϕ –g, . . . , Jk(tt0)

γuk(as1s0) + (1 –γ)ϕ –g

k

+sup

t∈G

1 +α(tt₀)

ξ^–1u₁(as₁s₀) –ϕ, . . . ,u_k(as₁s₀) –ϕ

k

–J1(tt0)u1(as1s0) –ϕ, . . . ,Jk(tt0)uk(as1s0) –ϕ

k+d·α(tt0) +sup

t∈G

1 +α(tt₀)γu₁(as₁s₀) + (1 –γ)ϕ–g, . . . ,γu_k(as₁s₀) + (1 –γ)ϕ–g

k

≤+ (1 –)sup

t∈Gξ^–1u1(as1s0) –ϕ, . . . ,uk(as1s0) –ϕ

k

–u1(tt0as1s0) –ϕ, . . . ,uk(tt0as1s0) –ϕ

k+ϕ(as1s0) +

+ (1 +)γu1(as1s0) + (1 –γ)ϕ–g, . . . ,γuk(as1s0) + (1 –γ)ϕ–g

k

≤+ (1 +)ξ^–1(5)

+ (1 +)γu1(as1s0) + (1 –γ)ϕ–g, . . . ,γuk(as1s0) + (1 –γ)ϕ–g

k. Then

s∈Ginfsup

t∈G

γu₁(ts) + (1 –γ)ϕ–g, . . . ,γu_k(ts) + (1 –γ)ϕ–g

k

≤(1 +)ξ^–1(5) + (1 +)inf

a∈Gγu₁(as₁s₀) + (1 –γ)ϕ–g, . . . ,γu_k(as₁s₀) + (1 –γ)ϕ–g

k

≤(1 +)ξ^–1(5) + (1 +)sup

b∈Ginf

a∈Gγu1(ab) + (1 –γ)ϕ–g, . . . ,γuk(ab) + (1 –γ)ϕ–g

k. Hence,

s∈Ginfsup

t∈G

γu1(ts) + (1 –γ)ϕ–g, . . . ,γuk(ts) + (1 –γ)ϕ–g

k

≤sup

s∈Ginf

t∈Gγu1(ts) + (1 –γ)ϕ–g, . . . ,γuk(ts) + (1 –γ)ϕ–g

k

because> 0 is arbitrary.

Theorem 2.5 Assume that{(X^k,·k)}k∈Nis a uniformly convex multi-Banach space,and suppose that∅ =C⊂X is bounded and closed.Assume thatk={J_k(t) :t∈G}for each k≥1is a reversible semigroup of asymptotically nonexpansive functions on C.If D has a left invariant mean,then there exists a retraction PkfromLAO(k)onto F(k)in which:

(1) Pkis nonexpansive in the sense (P1u1–P1v1, . . . ,Pkuk–Pkvk)

k

≤inf

s∈Gsup

t∈G

u1(st) –v1(st), . . . ,uk(st) –vk(st)

k, ∀uk,vk∈LAO(k);

(2) PkJk(p)uk=Jk(p)Pkuk=Pkukfor alluk∈AO(k)andp∈G;

(3) Pkuk∈

s∈Gconv{uk(t) :t≥s}for alluk∈LAO(k).

Proof We knowDhas a left invariant mean, so there is a net{γ_k,α:α∈A}of finite means onGin whichlim_α∈A(γ1,α–^∗_sγ_1,α, . . . ,γ_k,α–^∗_sγ_k,α)k= 0 for everys∈G, in whichAis a directed system. PuttingI=A×G={β= (α,t) :α∈A,t∈G}. Forβ_i= (α_i,t_i)∈I,i= 1, 2, defineβ₁≤β₂iffα₁≤α₂,t₁≤t₂. Then,Iis also a directed system. For eachβ= (α,t)∈I, definePk,1β=α,Pk,2β=t, andγβ=γα. So, for everys∈G,

limβ∈Iγ_1,β–^∗γ_1,β, . . . ,γ_k,β–^∗γ_k,β

k= 0. (2.1)

Assume thatγ ={{tβ}β∈I,tβ≥Pk,2β,∀β∈I}. Taking any{tβ,β∈I} ∈γ, sincer^∗_tβγ_k,βis bounded, without loss of generality, letr^∗_tβγ_k,β beweak^∗ convergent. Then, for alluk∈

LAO(k),ω-limβ∈Iγ_k,β(t)uk(ttβ)exist. We define P_ku_k=ω-lim

β∈Iγk,β(t)u_k(ttβ) .

On the other hand, for everyuk∈LAO(k),Pkuk∈

s∈Gconv{u(t) :t≥s}. Next, we shall show thatPkuk∈F(k). Then, for every∈(0, 1], there ist0∈Gsuch that, for eacht≥t0, ϕ(t) < ^a()₄ . Also, we can suppose thatP_k2β≥t₀for everyβ∈I, sotβ≥t₀,{tβ} ∈γ. From Lemma2.3, for everyp∈G_a(a(a()

16)), there isn∈Nsuch that, for eacht∈Gandβ∈I, 1

n n

i=1

Jk

pⁱ

uk(ttβ)∈Fa() 4

Jk(p) .

Since for everyt∈G

1

n n

i=1

J1

pⁱ

u1(ttβ) –1 n

n i=1

u1

pⁱttβ

, . . . ,1 n

n i=1

Jk

pⁱ

uk(ttβ) –1 n

n i=1

uk

pⁱttβ

k

≤ϕ(ttβ) <a() 4 , we have, for everyp∈G_a(a(a()

16)), 1

n n

i=1

u_k pⁱttβ

∈Fa() 4

J_k(p) +B

0,a()

4

⊂Fa()

J_k(p) .

Equation (2.1) implies that

limβ∈I

γ_1,β(t)

1 n

n i=1

u1

pⁱttβ

–γ_1,βu1(ttβ) , . . . ,

γ_k,β(t) 1

n n

i=1

u_k pⁱttβ

–γ_k,βu_k(ttβ)

k

= 0.

Combining it with the definition ofPkuk, we get, for allp∈G_a(a(a() 16)), Pkuk=ω-lim

β∈Iγ_k,β(t)

1 n

n i=1

uk

pⁱttβ

∈coFa()

Jk(p) .

Lemma2.1also implies that for everyp∈G_a(a(a()

16)),Pkuk∈F(Jk(p)). Now, the continuity ofJ_k(p) implies thatP_ku_k∈F(k). Obviously, for anyp∈G,

PkJk(p)uk=ω-lim

β∈Iγ_k,β(t)Jk(p)uk(ttβ)

=ω-lim

β∈Iγ_k,β(t)uk(httβ)

=ω-lim

β∈Iγ_k,β(t)uk(ttβ)

(using (2.1))

=Pkuk

and for everyvk∈LAO(k) ands∈G, we have (P₁u₁–P₁v₁, . . . ,P₁u₁–P₁v₁)

k

≤lim inf

β∈I γ_1,β(t)u1(ttβ)

–γ_1,β(t)v1(ttβ)

, . . . ,γk,β(t)uk(ttβ)

–γ_k,β(t)vk(ttβ)

k

=lim inf

β∈I γ_1,β(t)u1(sttβ)

–γ_1,β(t)v1(sttβ) , . . . , γ_k,β(t)u_k(sttβ)

–γ_k,β(t)v_k(sttβ)

k (by (2.1))

≤lim inf

β∈I γ_1,β(t), . . . ,γ_1,β(t)

k·sup

t∈G

u₁(sttβ) –v₁(sttβ), . . . ,u_k(sttβ) –v_k(sttβ)

k

≤sup

t∈G

u1(st) –v1(st), . . . ,uk(st) –vk(st)

k. Thus,

(P1u1–P1v1, . . . ,Pkuk–Pkvk)

k≤inf

s∈Gsup

t∈G

u1(st) –v1(st), . . . ,uk(st) –vk(st)

k.

Theorem 2.6(Ergodic theorem [17]) Assume that{(X^k, · k)}k∈Nis a uniformly convex multi-Banach space,and suppose that∅ =C⊂X is bounded and closed.Assume thatk= {Jk(t) :t∈G}is a reversible semigroup of asymptotically nonexpansive functions on C.If D has a left invariant mean and there is a unique retraction PkfromLAO(K)onto F(k), which satisfies properties(1)–(3)in Theorem2.5,then for every strongly net{ν_k,α:α∈A}

on D and uk∈AO(k), ω-lim

α∈A

uk(tp)dνk,α(t) =Pk∈F(k) uniformly in p∈γ(G), in whichγ(G) ={s∈G:st=ts for all t∈G}.

Theorem 2.7 Assume that{(X^k, · k)}k∈N is a uniformly convex multi-Banach space, and suppose that∅ =C⊂X is bounded and closed.Assume thatk ={Jk(t) :t∈G} of a reversible semigroup of asymptotically nonexpansive mappings on C,and let uk(·)be an almost orbit ofk.If

ω-lim

t∈Guk(pt) –uk(t) = 0 for every p∈G,then

ωω(u_k)⊂F(k).

Proof Let∈(0, 1], then there ist0∈Gsuch that, fort≥t0, ϕ(t) <^a()₄ . Suppose that pk∈ω_ω(uk), so we can find a subnet{uk(tα)}α∈Aof{uk(t)}t∈G withω-lim_α∈Auk(tα) =pk

in which, for everyα∈A,tα≥t₀, in whichAis a directed system. Using Lemma2.3, for everyp∈G_a(a(^a()

16)), we can findn∈Nsuch that, for everyα∈A, 1

n n

i=1

Jk

pⁱ

uk(tα)∈Fa() 4

Jk(p) .