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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

Journal of Inequalities and Applications20192019:43

https://doi.org/10.1186/s13660-019-1996-8

  • Received: 21 November 2018
  • Accepted: 11 February 2019
  • Published:

Abstract

Let Y be 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 in Y. 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.

Keywords

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

MSC

  • 39A10
  • 39B72
  • 47H10
  • 46B03

1 Preliminaries

Dales and Polyakov in [1] introduced a multi–normed space by using the concept of operator sequence space, operator spaces, and Banach lattices; for more details and application, we refer to [13].

In this paper assume that \(({Y},\|\cdot\|)\) is a complex normed space, and let \(\ell\in\mathbb{N}\). We denote by \({Y}^{\ell}\) the vector space \({Y}\oplus\cdots\oplus{Y}\) consisting of -tuples \((y_{1}, \ldots , y_{\ell})\), where \(y_{1}, \ldots , y_{\ell}\in{Y}\). The linear operations on \({Y}^{\ell}\) are defined coordinate-wise. The zero element of either Y or \({Y}^{\ell}\) is denoted by 0. We denote by \({\mathbb {N}}_{\ell}\) the set \(\{1, 2, \ldots ,\ell\}\) and by \(\varSigma_{\ell}\) the group of permutations on symbols.

Definition 1.1

Suppose that Y is a vector space, and take \(\ell\in\mathbb{N}\). For \(\sigma\in\varSigma_{\ell}\), define
$$B_{\sigma}(y) = (y_{\sigma(1)},\ldots,y_{\sigma(\ell)}),\quad y= (y_{1},\ldots,y_{\ell})\in Y^{\ell}. $$
For \(\beta= (\beta_{j}) \in{\mathbb {C}} ^{\ell}\), define
$$K_{\beta}(y) = (\beta_{j}y_{j}),\quad y= (y_{1},\ldots,y_{\ell})\in Y^{\ell}. $$

Definition 1.2

Assume that \((Y,\|\cdot\|)\) is a complex (respectively, real) normed space, and take \(m \in\mathbb{N}\). A multi-norm of level m on \(\{Y^{\ell}: \ell\in{\mathbb {N}}_{m}\}\) is a sequence \((\|\cdot\|_{\ell}:\ell \in{\mathbb {N}}_{m})\) such that \(\|\cdot\|\) is a norm on \(Y^{\ell}\) for each \(\ell\in{\mathbb {N}}_{m}\), such that \(\|y\|_{1} = \|y\|\) for each \(y \in Y\) (so that \(\|\cdot\|_{1}\) is the initial norm), and such that the following Axioms (a1)–(a4) are satisfied for each \(\ell \in{\mathbb {N}}_{m}\) with \(k\geq2\):
  1. (a1)
    for each \(\sigma\in\varSigma_{\ell}\) and \(y\in Y^{\ell}\), we have
    $$\bigl\Vert B_{\sigma}(y) \bigr\Vert _{\ell}= \Vert y \Vert _{\ell}; $$
     
  2. (a2)
    for each \(\beta_{1},\ldots,\beta_{\ell}\in{\mathbb {C}}\) (respectively, each \(\beta_{1},\ldots,\beta_{\ell}\in{\mathbb {R}}\)) and \(y\in Y^{\ell}\), we have
    $$\bigl\Vert K_{\beta}(y) \bigr\Vert _{\ell}\leq\Bigl(\max _{j\in{\mathbb {N}}_{\ell}} \vert \beta_{j} \vert \Bigr) \Vert y \Vert _{\ell}; $$
     
  3. (a3)
    for each \(y_{1},\ldots,y_{\ell-1}\), we have
    $$\bigl\Vert (y_{1},\ldots,y_{\ell-1},0) \bigr\Vert _{\ell}= \bigl\Vert (y_{1},\ldots,y_{\ell-1}) \bigr\Vert _{\ell-1}; $$
     
  4. (a4)
    for each \(y_{1},\ldots,y_{\ell-1}\in Y \),
    $$\bigl\Vert (y_{1},\ldots,y_{\ell-2},y_{\ell-1},y_{\ell-1}) \bigr\Vert _{\ell}= \bigl\Vert (y_{1},\ldots,y_{\ell -1}) \bigr\Vert _{\ell-1}. $$
     

In this case, \((({Y}^{\ell},\|\cdot\|_{\ell}): \ell\in{\mathbb {N}}_{m})\) is a multi-normed space of level m.

A multi-norm on \(\{Y^{\ell}: \ell\in{\mathbb {N}}\}\) is a sequence
$$\bigl(\|\cdot\|_{\ell}\bigr)=\bigl(\|\cdot\|_{\ell}: \ell\in{\mathbb {N}}\bigr) $$
such that \((\|\cdot\|_{\ell}: \ell\in{\mathbb {N}}_{m})\) is a multi-norm of level m for each \(m\in{\mathbb {N}} \). In this case, \((({Y}^{m},{\|\cdot\|} _{m}): m\in{\mathbb {N}})\) is a multi-normed space.

Lemma 1.3

([3])

Let \((({Y}^{\ell},\|\cdot\|_{\ell}): \ell\in\mathbb{N})\) be a multi-normed space, and take \(\ell\in{\mathbb {N}}_{m}\). Then
  1. (a)

    \(\|(y,\ldots,y)\|_{\ell}=\|y\|\) (\(y\in Y\));

     
  2. (b)

    \(\max_{j\in{\mathbb {N}}_{\ell}}\|y_{j}\|\leq \|(y_{1},\ldots,y_{\ell})\|_{\ell}\leq\sum_{j=1}^{\ell}\|y_{j}\|\leq\ell\max_{j\in {\mathbb {N}}_{\ell}}\|y_{j}\|\) (\(y_{1}, \ldots , y_{\ell}\in{Y}\)).

     

It follows from (b) that if \(({Y},\|\cdot\|)\) is a Banach space, then \(( {Y}^{\ell},\|\cdot\|_{\ell})\) is a Banach space for each \(\ell\in\mathbb{N}\); in this case \((({Y}^{\ell},\|\cdot\|_{\ell}): \ell\in\mathbb{N})\) is a multi-Banach space.

Example 1.4

([1])

The sequence \((\|\cdot\|_{\ell}: \ell\in\mathbb{N})\) on \(\{{Y}^{\ell}: \ell\in\mathbb{N}\}\) defined by
$$\bigl\Vert (y_{1},\ldots,y_{\ell}) \bigr\Vert _{\ell}:=\max_{j\in{\mathbb {N}}_{\ell}} \Vert y_{j} \Vert \quad (y_{1}, \ldots , y_{\ell}\in {Y}) $$
is a multi-norm called the minimum multi-norm.

Example 1.5

([1])

Assume that \(\{(\|\cdot\|_{\ell}^{\beta}: \ell\in\mathbb{N}):\beta\in B\}\) is the (non-empty) family of all multi-norms on \(\{{Y}^{\ell}:\ell\in\mathbb{N}\}\). For \(\ell\in\mathbb{N} \), set
$$\bigl\Vert (y_{1},\ldots,y_{\ell}) \bigr\Vert _{k}:=\sup_{\beta\in B} \bigl\Vert (y_{1},\ldots y_{\ell}) \bigr\Vert _{\ell}^{\beta}\quad (y_{1}, \ldots , y_{\ell}\in {Y}). $$
Then \(( \|\cdot\|_{\ell}: \ell\in\mathbb{N})\) is a multi-norm on \(\{{Y}^{\ell}: \ell\in\mathbb{N}\}\), called the maximum multi-norm.
By the property (b) of multi-norms and the triangle inequality for the norm \(\|\cdot\|_{k}\), we can get the following properties. Suppose that \((({Y}^{\ell},\|\cdot\|_{\ell}): \ell\in\mathbb{N})\) is a multi-normed space. Let \(\ell\in\mathbb{N}\) and \((y_{1}, \ldots , y_{\ell})\in {Y}^{k} \). For every \(i\in\{1,\ldots,\ell\}\), let \((y_{m}^{i})_{m=1,2,\ldots}\) be a sequence in Y such that \(\lim_{m\to\infty}y_{m}^{i}=y_{i}\). Then for each \((z_{1},\ldots,z_{\ell})\in{Y}^{\ell}\) we have
$$\lim_{m\to \infty}\bigl(y_{m}^{1}-z_{1}, \ldots,y_{m}^{\ell}-z_{\ell}\bigr)=(y_{1}-z_{1}, \ldots,y_{\ell}-z_{\ell}). $$
A sequence \((y_{m})\) in Y is a multi-null sequence if, for every \(\varepsilon>0\), there exists \(m_{0}\in\mathbb{N}\) such that
$$\sup_{\ell\in\mathbb{N}} \bigl\Vert (y_{n},\ldots,y_{m+\ell-1}) \bigr\Vert _{\ell}< \varepsilon \quad (m\geq m_{0}). $$
Let \(y\in Y\). We say that the sequence \((y_{m})\) is multi-convergent to \(y\in{Y}\) and write
$$\lim_{m\to\infty}y_{m}=y $$
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 space X. Let \(X^{*}\) be the dual of X, then the value of \(u^{*}\in X^{*}\) at \(u \in X\) will be denoted by \(\langle u, u^{*}\rangle \), and we associate the set
$$J(u) =\bigl\{ u^{*}\in X : \bigl\langle u, u^{*}\bigr\rangle =\|u\|^{2}= \bigl\Vert u^{*} \bigr\Vert ^{2}\bigr\} . $$
It is clear from the Hahn–Banach theorem that \(J(u)\) is not empty for all \(u \in X\). Then the multi-valued operator \(J : X\to X^{*}\) is called the normalized duality mapping of X, also \(\Im_{k} = \{ J_{k}(t) : t \in G \}\) is a reversible semigroup of asymptotically nonexpansive functions acting on C. Let \(F(\Im_{k})\) denote the set of all fixed points of \(\Im_{k}\), i.e., \(F(\Im_{k}) = \{ u \in C : J_{k}(t)u=u, \forall t\in G \}\). For each \(\epsilon>0\) and \(p\in G\), we put
$$F_{\epsilon}\bigl(J_{k}(p)\bigr)=\bigl\{ u\in C : \bigl\Vert \bigl(J_{1}(p)u - u,\ldots,J_{k}(p)u - u\bigr) \bigr\Vert _{k}\leq \epsilon\bigr\} . $$
Note that if, for any \(\epsilon>0\), there exists \(p_{\epsilon}\in G\) such that for all \(p>p_{\epsilon}\), \(u \in F_{\epsilon}(J_{k}(p))\), then \(\lim_{p\in G} J_{k}(p)u = u\); moreover, \(u\in F(\Im_{k})\) by the continuity of elements \(\{J_{k}(p),p\in G\}\) (for more details, we refer to [49]).

We denote the set of all almost orbits of \(\Im_{k}\) and the set \(\{ J_{k}(p)u_{k}(\cdot) : p\in G, u_{k} \in \operatorname{AO}(\Im_{k})\}\) by \(\operatorname{AO}(\Im_{k})\) and \(\operatorname{LAO}(\Im _{k})\), respectively. Denote by \(\omega_{\omega}(u_{k})\) the set of all weak limit points of subnets of net \(\{u_{k}(t)\}_{t\in G}\).

Lemma 1.6

([10])

Assume that X is a Banach space and J is the normalized duality function. Therefore
$$\|u+v\|^{2} \leq\|u\|^{2} + 2\bigl\langle v,j(u+v)\bigr\rangle $$
for all \(j(u+v)\in J(u+v)\) and \(u,v\in X\).

Lemma 1.7

([11])

Assume that \(\{(X^{k},\|\cdot\|_{k})\}_{k\in\mathbb{N}}\) is a uniformly convex multi-Banach space and \(\emptyset\neq C\subset X^{k}\) is a bounded closed convex set. Then there exists a strictly increasing continuous convex function \(\xi: [0,+\infty)\rightarrow[0,+\infty)\) with \(\xi(0) =0\) such that
$$\begin{aligned}& \xi \Biggl( \Biggl\Vert \Biggl(J_{1}\Biggl(\sum _{i=1}^{n} a_{i}u_{i}\Biggr)-\sum _{i=1}^{n} a_{i}J_{1}u_{i}, \ldots,J_{k}\Biggl(\sum_{i=1}^{n} a_{i}u_{i}\Biggr)-\sum_{i=1}^{n} a_{i}J_{k}u_{i}\Biggr) \Biggr\Vert _{k} \Biggr) \\& \quad \leq\max_{1\leq i,j\leq n}\bigl\{ \Vert u_{i}-u_{j} \Vert - \bigl\Vert (J_{1}u_{i} -J_{1}u_{j}, \ldots,J_{k}u_{i} -J_{k}u_{j}) \bigr\Vert _{k}\bigr\} \end{aligned}$$
for all integers \(a_{1},\ldots,a_{n} \geq0\), \(n\geq1\) with \(\sum_{i=1}^{n} a_{i} =1\), \(u_{1},\ldots,u_{n} \in C\), and every nonexpansive function \(J_{k}\) of C to C.
Lemma 1.7 implies that, for all \(a_{1},\ldots,a_{n} \geq0\) with \(\sum_{i=1}^{n} a_{i} =1\), \(u_{1},\ldots,u_{n} \in C\),
$$\begin{aligned}& \Biggl\Vert \Biggl(J_{1}(p) \Biggl(\sum _{i=1}^{n} a_{i}u_{i}\Biggr)- \sum _{i=1}^{n} a_{i}J_{1}(p)u_{i}, \ldots,J_{k}(p) \Biggl(\sum_{i=1}^{n} a_{i}u_{i}\Biggr)- \sum_{i=1}^{n} a_{i}J_{k}(p)u_{i}\Biggr) \Biggr\Vert _{k} \\& \quad \leq \bigl(1 + \alpha(p)\bigr)\xi^{-1} \biggl( \max _{1\leq i,j\leq n} \biggl\{ \Vert u_{i} - u_{j} \Vert \\& \qquad {}- \frac{1}{1 + \alpha(p)} \bigl\Vert \bigl(J_{1}(p)u_{i} -J_{1}(p)u_{j},\ldots,J_{k}(p)u_{i} -J_{k}(p)u_{j}\bigr) \bigr\Vert _{k}\biggr\} \biggr) \\& \quad \leq\bigl(1 + \alpha(p)\bigr)\xi^{-1} \Bigl( \max _{1\leq i,j\leq n} \bigl\{ \Vert u_{i} - u_{j} \Vert \\& \qquad {}- \bigl\Vert \bigl(J_{1}(p)u_{i} -J_{1}(p)u_{j},\ldots,J_{k}(p)u_{i} -J_{k}(p)u_{j}\bigr) \bigr\Vert _{k}\bigr\} + d \cdot\alpha(p) \Bigr) \end{aligned}$$
in which \(d = 4\sup\{\|u\| : u \in C\}+1\).
For every \(\epsilon\in(0,1]\), define
$$a(\epsilon) = \min \biggl\{ \frac{{\epsilon}^{2}}{(d +2)^{2}},\frac{{\epsilon }^{3}}{(3d +2)^{2}}\xi\biggl( \frac{\epsilon}{4}\biggr) \biggr\} $$
and
$$G_{\epsilon} = \bigl\{ h \in G : \alpha(p) \leq\epsilon\bigr\} , $$
in which \(\xi(\cdot) \) is as Lemma 1.7. Then \(G_{\epsilon}\neq \emptyset\) for \(\epsilon> 0\), and if \(p\in G_{\epsilon}\), then for all \(t \geq p\), \(t\in G_{\epsilon}\). Note that \(G_{a({\epsilon})} \subset G_{\epsilon }\) for all \(\epsilon\in(0,1]\).

2 Main result

For studies on ergodic theory and its history, we refer to [430]. The results of this paper are an extension and generalization of [31].

Lemma 2.1

For all \(p\in G_{a({\epsilon})}\),
$$\overline{\operatorname{co}}\,F_{a({\epsilon})}\bigl(J_{k}(p)\bigr) \subset F_{\epsilon}\bigl(J_{K}(p)\bigr). $$

Proof

Since \(F_{\epsilon}(J_{K}(p))\) is closed, we only need to prove that, for all \(p\in G_{a({\epsilon})}\),
$$\operatorname{co}F_{a({\epsilon})}\bigl(J_{k}(p)\bigr)\subset F_{\epsilon}\bigl(J_{K}(p)\bigr). $$
Let \(v=\sum_{i=1}^{n} a_{i}v_{i}\), \(v_{i} \in F_{a({\epsilon})}(J_{k}(p))\), \(a_{i} \geq 0\), \(i=1,\ldots,n\), and \(\sum_{i=1}^{n} a_{i} = 1\). Then
$$\begin{aligned}& \bigl\Vert \bigl(J_{1}(p)v-v,\ldots,J_{k}(p)v-v\bigr) \bigr\Vert _{k} \\& \quad = \Biggl\Vert \Biggl(J_{1}(p)\sum _{i=1}^{n} a_{i}v_{i} - \sum _{i=1}^{n} a_{i}v_{i}, \ldots,J_{k}(p)\sum_{i=1}^{n} a_{i}v_{i} - \sum_{i=1}^{n} a_{i}v_{i}\Biggr) \Biggr\Vert _{k} \\& \quad \leq \Biggl\Vert \Biggl(J_{1}(p)\sum _{i=1}^{n} a_{i}v_{i} - \sum _{i=1}^{n} a_{i}J_{1}(p)v_{i}, \ldots,J_{k}(p)\sum_{i=1}^{n} a_{i}v_{i} - \sum_{i=1}^{n} a_{i}J_{k}(p)v_{i}\Biggr) \Biggr\Vert _{k} \\& \quad \leq2\xi^{-1} \Bigl(\max_{1\leq i,j\leq n}\bigl\{ \Vert v_{i}-v_{j} \Vert - \bigl\Vert \bigl(J_{1}(p)v_{i}-J_{1}(p)v_{i}, \ldots,J_{k}(p)v_{i}-J_{k}(p)v_{i}\bigr) \bigr\Vert _{k}\bigr\} + d \cdot\alpha (p) \Bigr) \\& \qquad {}+ a(\epsilon) \\& \quad \leq2\xi^{-1} \Bigl(\max_{1\leq i,j\leq n}\bigl\{ \bigl\Vert \bigl(v_{i}-J_{1}(p)v_{i}, \ldots,v_{i}-J_{k}(p)v_{i}\bigr) \bigr\Vert _{k} + \bigl\Vert \bigl(v_{j}-J_{1}(p)v_{j}, \ldots,v_{j}-J_{k}(p)v_{j}\bigr) \bigr\Vert _{k}\bigr\} \\& \qquad {}+ d \cdot\alpha(p) \Bigr)+ a(\epsilon) \\& \quad \leq2\xi^{-1} \bigl( 2a(\epsilon) + d \cdot a(\epsilon) \bigr) + a(\epsilon) \\& \quad \leq\frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon. \end{aligned}$$
 □

Lemma 2.2

For every \(p\in G_{\frac{\epsilon}{4}}\),
$$F_{\frac{\epsilon}{4}}\bigl(J_{k}(p)\bigr) + B\biggl(0,\frac{\epsilon}{4} \biggr)\subset F_{\epsilon}\bigl(J_{k}(p)\bigr). $$

Proof

Let \(p \in G_{\frac{\epsilon}{4}}\) and \(u = v + w \in F_{\frac{\epsilon }{4}}(J_{k}(p)) + B(0,\frac{\epsilon}{4})\), where \(v\in F_{\frac{\epsilon }{4}}(J_{k}(p)) \) and \(w \in B(0,\frac{\epsilon}{4})\), then
$$\begin{aligned}& \bigl\Vert \bigl(J_{1}(p)u - u,\ldots,J_{k}(p)u - u\bigr) \bigr\Vert _{k} \\& \quad = \bigl\Vert \bigl(J_{1}(p) (v+w)-(v+w),\ldots,J_{k}(p) (v+w)-(v+w)\bigr) \bigr\Vert _{k} \\& \quad \leq \bigl\Vert \bigl(J_{1}(p) (v+w) - J_{1}(p)v, \ldots,J_{k}(p) (v+w) - J_{k}(p)v\bigr) \bigr\Vert _{k} \\& \qquad {}+ \bigl\Vert \bigl(J_{1}(p)v - v,\ldots,J_{k}(p)v - v\bigr) \bigr\Vert _{k}+ \Vert w \Vert \\& \quad \leq2 \Vert w \Vert + \bigl\Vert \bigl(J_{1}(p)v - v, \ldots,J_{k}(p)v - v\bigr) \bigr\Vert _{k} + \Vert w \Vert \\& \quad \leq3\frac{\epsilon}{4}+ \frac{\epsilon}{4} = \epsilon. \end{aligned}$$
 □

Lemma 2.3

Assume that \(\epsilon\in(0,1]\) and \(p\in G_{a(a(\frac{\epsilon }{4}))}\), so we can find \(n_{0}\in N\) such that, for each \(n\geq n_{0}\) and \(u\in C\),
$$\frac{1}{n} \sum_{i=1}^{n}J_{k} \bigl(p^{i}\bigr)u \in F_{\epsilon}\bigl(J_{k}(p)\bigr). $$

Proof

Let \(\epsilon\in(0,1]\) and \(m= \frac{2d+1}{a(\frac{\epsilon}{4})}\). There is \(n_{0}\in N\) satisfying
$$n_{0} \geq\max \biggl\{ \frac{12md}{\epsilon},32m^{2}d(d+1) \biggl(\xi \biggl( \frac{a(\frac{\epsilon}{4})}{2} \biggr)\epsilon \biggr)^{-1} \biggr\} . $$
For any \(n\geq n_{0}\) and \(p\in G_{a(a(\frac{\epsilon}{4}))}\), we can take a number
$$K = m^{2}d\bigl(1+2n\alpha(p)\bigr) \biggl(\xi \biggl( \frac{a(\frac{\epsilon }{4})}{2} \biggr) \biggr)^{-1} \quad \biggl( k< \frac{n}{2} \biggr). $$
For every \(i\in N\) and \(u\in C\), we put
$$\begin{aligned} a_{i}(u) =&\xi\Biggl(\frac{8}{9} \Biggl\Vert \Biggl(\frac{1}{m}\sum _{j=1}^{m} J_{1} \bigl(p^{i+j+1}\bigr)u - J_{1}(p)\frac{1}{m}\sum _{j=1}^{m} J_{1}\bigl(p^{i+j} \bigr)u,\ldots, \\ & \frac{1}{m}\sum_{j=1}^{m} J_{k}\bigl(p^{i+j+1}\bigr)u - J_{k}(p) \frac{1}{m}\sum_{j=1}^{m} J_{k}\bigl(p^{i+j}\bigr)u\Biggr) \Biggr\Vert _{k} \Biggr). \end{aligned}$$
By \(\alpha(p)\leq\frac{1}{8}\) and
$$\begin{aligned} a_{i}(u) \leq& \max_{1\leq j,t \leq m}\bigl\{ \bigl\Vert J_{1}\bigl(p^{i+j}\bigr)u-J_{k}\bigl(p^{i+t} \bigr)u,\ldots,J_{k}\bigl(p^{i+j}\bigr)u-J_{k} \bigl(p^{i+t}\bigr)u \bigr\Vert _{k} \\ &{}- \bigl\Vert J_{1}\bigl(p^{i+j+1}\bigr)u-J_{k} \bigl(p^{i+t+1}\bigr)u,\ldots,J_{k}\bigl(p^{i+j+1} \bigr)u-J_{k}\bigl(p^{i+t+1}\bigr)u \bigr\Vert _{k} + d \cdot\alpha(p)\bigr\} \\ \leq& \sum_{1\leq j< t \leq m} \bigl( \bigl\Vert \bigl(J_{1}\bigl(p^{i+j}\bigr)u-J_{1} \bigl(p^{i+t}\bigr)u,\ldots J_{k}\bigl(p^{i+j} \bigr)u-J_{k}\bigl(p^{i+t}\bigr)u\bigr) \bigr\Vert _{k} \\ &{}- \bigl\Vert \bigl(J_{1}\bigl(p^{i+j+1} \bigr)u-J_{1}\bigl(p^{i+t+1}\bigr)u,\ldots J_{k} \bigl(p^{i+j+1}\bigr)u-J_{k}\bigl(p^{i+t+1}\bigr)u\bigr) \bigr\Vert _{k} + d \alpha(p) \bigr), \end{aligned}$$
we get
$$\begin{aligned}& \sum_{i=1}^{n} a_{i}(u) \\& \quad \leq\sum_{i=1}^{n} \sum _{1\leq j< t \leq m} \bigl( \bigl\Vert J_{1}\bigl(p^{i+j} \bigr)u-J_{k}\bigl(p^{i+t}\bigr)u,\ldots,J_{k} \bigl(p^{i+j}\bigr)u-J_{k}\bigl(p^{i+t}\bigr)u \bigr\Vert _{k} \\& \qquad {}- \bigl\Vert J_{1}\bigl(p^{i+j+1} \bigr)u-J_{k}\bigl(p^{i+t+1}\bigr)u,\ldots,J_{k} \bigl(p^{i+j+1}\bigr)u-J_{k}\bigl(p^{i+t+1}\bigr)u \bigr\Vert _{k} + d \cdot\alpha(p) \bigr) \\& \quad = \sum_{1\leq j< t \leq m} \sum _{i=1}^{n} \bigl( \bigl\Vert J_{1} \bigl(p^{i+j}\bigr)u-J_{k}\bigl(p^{i+t}\bigr)u, \ldots,J_{k}\bigl(p^{i+j}\bigr)u-J_{k} \bigl(p^{i+t}\bigr)u \bigr\Vert _{k} \\& \qquad {}- \bigl\Vert J_{1}\bigl(p^{i+j+1} \bigr)u-J_{k}\bigl(p^{i+t+1}\bigr)u,\ldots,J_{k} \bigl(p^{i+j+1}\bigr)u-J_{k}\bigl(p^{i+t+1}\bigr)u \bigr\Vert _{k} + d \cdot\alpha(p) \bigr) \\& \quad \leq\sum_{1\leq j< t \leq m} \bigl(d + nd \cdot\alpha(p) \bigr) \leq m^{2}d\bigl(1+n\alpha(p)\bigr). \end{aligned}$$
Suppose that there is an element say t in \(\{a_{i}(u) : i = 1,2,\ldots,2n\} \) such that if \(a_{i}(u)\geq\xi(\frac{a(\frac{\epsilon}{4})}{2})\), then
$$t\xi\biggl(\frac{a(\frac{\epsilon}{4})}{2}\biggr)\leq m^{2}d\bigl(1+2n\alpha(p) \bigr). $$
Hence
$$t\leq m^{2}d\bigl(1+2n\alpha(p)\bigr) \biggl(\xi\biggl( \frac{a(\frac{\epsilon}{4})}{2}\biggr) \biggr)^{-1} = K. $$
So, there are at most \(N=[K]\) terms in \(\{a_{i}(u) : i = 1,2,\ldots,2n\}\) with \(a_{i}(u)\geq\xi(\frac{a(\frac{\epsilon}{4})}{2})\). Then, for every i in \(\{1,2,\ldots,n\}\), there exists at least one term \(a_{i+j_{0}}(u)\) (\(0\leq j_{0}\leq N\)) in \(\{a_{i+j}(u) : j = 0,1,\ldots,N\}\) hold \(a_{i+j_{0}} < \xi(\frac{a(\frac{\epsilon}{4})}{2})\).
Put
$$\ell_{i} = \min\biggl\{ j : a_{i+j}(u)< \xi\biggl( \frac{a(\frac{\epsilon}{4})}{2}\biggr), 0\leq j \leq N \biggr\} , $$
\(i=1,2,\ldots,n\). Now, there are at most N elements in \(\{i=1,2,\ldots,n\}\) such that \(\ell_{i} \neq0 \). Since
$$\begin{aligned}& \Biggl\Vert \Biggl( J_{1}(p)\frac{1}{m}\sum _{j=1}^{m} J_{1}\bigl(p^{i+\ell_{i}+j} \bigr)u - \frac{1}{m}\sum_{j=1}^{m} J_{1}\bigl(p^{i+\ell_{i}+j}\bigr)u,\ldots, \\& \qquad J_{k}(p) \frac{1}{m}\sum_{j=1}^{m} J_{k}\bigl(p^{i+\ell_{i}+j}\bigr)u - \frac{1}{m}\sum _{j=1}^{m} J_{k}\bigl(p^{i+\ell _{i}+j} \bigr)u \Biggr) \Biggr\Vert _{k} \\& \quad \leq \Biggl\Vert \Biggl( J_{1}(p)\frac{1}{m}\sum _{j=1}^{m} J_{1} \bigl(p^{i+\ell _{i}+j}\bigr)u - \frac{1}{m}\sum _{j=1}^{m} J_{1}\bigl(p^{i+\ell_{i}+j+1} \bigr)u,\ldots, \\& \qquad J_{k}(p)\frac{1}{m}\sum_{j=1}^{m} J_{k}\bigl(p^{i+\ell_{i}+j}\bigr)u - \frac {1}{m}\sum _{j=1}^{m} J_{k}\bigl(h^{i+\ell_{i}+j+1} \bigr)u\Biggr) \Biggr\Vert _{k} \\& \qquad {}+ \Biggl\Vert \Biggl(\frac{1}{m}\sum _{j=1}^{m} J_{1}\bigl(p^{i+\ell_{i}+j} \bigr)u - \frac {1}{m}\sum_{j=1}^{m} J_{1}\bigl(p^{i+\ell_{i}+j+1}\bigr)u,\ldots, \\& \qquad \frac{1}{m}\sum _{j=1}^{m} J_{k} \bigl(p^{i+\ell_{i}+j}\bigr)u - \frac{1}{m}\sum _{j=1}^{m} J_{k}\bigl(p^{i+\ell _{i}+j+1} \bigr)u\Biggr) \Biggr\Vert _{k} \\& \quad \leq\frac{9}{8}\xi^{-1} \bigl(a_{i+\ell_{i}}(u)\bigr) + \frac{d}{2m} \\& \quad \leq\frac{9}{16}a\biggl(\frac{\epsilon}{4}\biggr)+\frac{1}{4}a \biggl(\frac{\epsilon }{4}\biggr)< a\biggl(\frac{\epsilon}{4}\biggr), \end{aligned}$$
we can conclude that, for all \(p \in G_{a(a(\frac{\epsilon}{4}))}\),
$$\frac{1}{m}\sum_{j=1}^{m} J_{k}\bigl(p^{i+\ell_{i}+j}\bigr)u\in F_{a(\frac{\epsilon }{4})} \bigl(J_{k}(p)\bigr). $$
By Lemma 2.1, we get, for all \(p \in G_{a(a(\frac{\epsilon }{4}))} \subset G_{a(\frac{\epsilon}{4})}\),
$$\frac{1}{n}\sum_{i=1}^{n} \frac{1}{m}\sum_{j=1}^{m} J_{k}\bigl(p^{i+\ell_{i}+j}\bigr)u \in \operatorname{co}F_{a(\frac{\epsilon}{4})} \bigl(J_{k}(p)\bigr)\subset F_{\frac{\epsilon}{4}}\bigl(J_{k}(p) \bigr). $$
Using Lemma 2.2 and
$$\begin{aligned}& \Biggl\Vert \Biggl(\frac{1}{n}\sum_{i=1}^{n}J_{1} \bigl(p^{i}\bigr)u - \frac{1}{n}\sum_{i=1}^{n} \frac{1}{m}\sum_{j=1}^{m} J_{1}\bigl(p^{i+\ell_{i}+j}\bigr)u,\ldots, \\& \qquad \frac{1}{n}\sum _{i=1}^{n}J_{k} \bigl(p^{i}\bigr)u - \frac{1}{n}\sum_{i=1}^{n} \frac{1}{m}\sum_{j=1}^{m} J_{k}\bigl(p^{i+\ell_{i}+j}\bigr)u\Biggr) \Biggr\Vert _{k} \\& \quad \leq\frac{1}{mn}\sum_{j=1}^{m} \Biggl\Vert \Biggl(\sum_{i=1}^{n}J_{1} \bigl(p^{i}\bigr)u - \sum_{i=1}^{n}J_{1} \bigl(p^{i+\ell_{i}+j}\bigr)u,\ldots,\sum_{i=1}^{n}J_{k} \bigl(p^{i}\bigr)u -sum_{i=1}^{n}J_{k} \bigl(p^{i+\ell_{i}+j}\bigr)u\Biggr) \Biggr\Vert _{k} \\& \quad \leq\frac{1}{mn}\sum_{j=1}^{m} \Biggl\Vert \Biggl(\sum_{i=1}^{n}J_{1} \bigl(p^{i}\bigr)u - \sum_{i=1}^{n}J_{1} \bigl(p^{i+j}\bigr)u,\ldots,\sum_{i=1}^{n}J_{k} \bigl(p^{i}\bigr)u - \sum_{i=1}^{n}J_{k} \bigl(p^{i+j}\bigr)u\Biggr) \Biggr\Vert _{k} \\& \qquad {}+ \frac{1}{mn}\sum_{j=1}^{m} \Biggl\Vert \Biggl(\sum_{i=1}^{n}L_{1} \bigl(p^{i+j}\bigr)u - \sum_{i=1}^{n}J_{1} \bigl(p^{i+\ell_{i}+j}\bigr)u,\ldots, \\& \qquad \sum_{i=1}^{n}J_{k} \bigl(p^{i+j}\bigr)u - \sum_{i=1}^{n}J_{k} \bigl(p^{i+\ell_{i}+j}\bigr)u\Biggr) \Biggr\Vert _{k} \\& \quad \leq\frac{md}{n}+\frac{Nd}{n} \\& \quad \leq\frac{\epsilon}{12}+\frac{m^{2}d^{2}(\xi(\frac{a(\frac{\epsilon }{4})}{2}))^{-1}}{n}+2m^{2}d^{2} \alpha(p) \biggl(\xi\biggl(\frac{a(\frac{\epsilon}{4})}{2}\biggr)\biggr)^{-1} \\& \quad < \frac{\epsilon}{12}+\frac{\epsilon}{32}+\frac{\epsilon}{8}< \frac {\epsilon}{4}, \end{aligned}$$
we obtain
$$\frac{1}{n}\sum_{i=1}^{n}J_{k} \bigl(p^{i}\bigr)u \in F_{\frac{\epsilon }{4}}\bigl(J_{k}(p)\bigr)+B \biggl(0,\frac{\epsilon}{4}\biggr)\subset F_{\epsilon}\bigl(J_{k}(p) \bigr). $$
 □

Lemma 2.4

Suppose that \(u_{k}(\cdot)\) is an almost orbit of \(\Im_{k}\). So
$$\lim_{t\in G} \bigl\Vert \bigl(\gamma u_{1}(t)+(1- \gamma)\varphi-g,\ldots,\gamma u_{k}(t)+(1-\gamma)\varphi-g\bigr) \bigr\Vert _{k} $$
exist for every \(\gamma\in(0,1)\) and \(\varphi,g\in F(\Im_{k})\).

Proof

To complete the proof, it is enough to prove that
$$\begin{aligned}& \inf_{s\in G}\sup_{t\in G} \bigl\Vert \bigl( \gamma u_{1}(ts)+(1-\gamma)\varphi -g,\ldots,\gamma u_{k}(ts)+(1- \gamma)\varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\sup_{s\in G}\inf_{t\in G} \bigl\Vert \bigl(\gamma u_{1}(ts)+(1-\gamma)\varphi -g,\ldots,\gamma u_{k}(ts)+(1-\gamma)\varphi-g\bigr) \bigr\Vert _{k}. \end{aligned}$$
We know, for every \(\epsilon>0\), there are \(t_{0}\) and \(s_{0}\in G\) such that, for any \(t\in G\), \(\alpha(tt_{0})<\frac{\epsilon}{1+d}\) and \(\varphi (ts_{0})<\epsilon\), where \(\varphi(t) = \sup_{p\in G} \| (u_{1}(pt)-J_{1}(p)u_{1}(t),\ldots,u_{k}(pt)-J_{k}(p)u_{k}(t))\|_{k}\). So, for every \(a\in G\),
$$\begin{aligned}& \inf_{s\in G}\sup_{t\in G} \bigl\Vert \bigl(u_{1}(tss_{0})-\varphi,\ldots,u_{k}(tss_{0})- \varphi \bigr) \bigr\Vert _{k} \\& \quad \leq\sup_{t\in G} \bigl\Vert \bigl(u_{1}(tt_{0}as_{0})- \varphi ,\ldots,u_{k}(tt_{0}as_{0})-\varphi\bigr) \bigr\Vert _{k} \\& \quad \leq\sup_{t\in G} \bigl\Vert \bigl(u_{1}(tt_{0}as_{0})-J_{1}(tt_{0})u_{1}(as_{0}), \ldots,u_{k}(tt_{0}as_{0})-J_{k}(tt_{0})u_{k}(as_{0}) \bigr) \bigr\Vert _{k} \\& \qquad {}+ \sup_{t\in G} \bigl\Vert J_{1}(tt_{0})u_{1}(as_{0})- \varphi ,\ldots,J_{k}(tt_{0})u_{k}(as_{0})- \varphi \bigr\Vert _{k} \\& \quad \leq\varphi(as_{0})+\sup_{t\in G}\bigl(1+ \alpha(tt_{0})\bigr)\cdot \bigl\Vert \bigl(u_{1}(as_{0})- \varphi,\ldots,u_{k}(as_{0})-\varphi\bigr) \bigr\Vert _{k} \\& \quad \leq \bigl\Vert \bigl(u_{1}(as_{0})-\varphi, \ldots,u_{k}(as_{0})-\varphi\bigr) \bigr\Vert _{k} +2\epsilon. \end{aligned}$$
Hence
$$\inf_{s\in G}\sup_{t\in G} \bigl\Vert \bigl(u_{1}(tss_{0})-\varphi,\ldots,u_{k}(tss_{0})- \varphi \bigr) \bigr\Vert _{k}\leq\inf_{a\in G} \bigl\Vert \bigl(u_{1}(as_{0})-\varphi,\ldots,u_{k}(as_{0})- \varphi\bigr) \bigr\Vert _{k}+2\epsilon. $$
Thus, there exists \(s_{1}\in G\) such that
$$\sup_{t\in G} \bigl\Vert \bigl(u_{1}(ts_{1}s_{0})- \varphi,\ldots,u_{k}(ts_{1}s_{0})-\varphi\bigr) \bigr\Vert _{k} < \inf_{a\in G} \bigl\Vert \bigl(u_{1}(as_{0})-\varphi,\ldots,u_{k}(as_{0})- \varphi\bigr) \bigr\Vert _{k}+3\epsilon. $$
Then, for every \(a\in G\), we get
$$\begin{aligned}& \inf_{s\in G}\sup_{t\in G} \bigl\Vert \bigl( \gamma u_{1}(ts)+(1-\gamma)\varphi -g,\ldots,\gamma u_{k}(ts)+(1- \gamma)\varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\sup_{t\in G} \bigl\Vert \bigl(\gamma u_{1}(tt_{0}as_{1}s_{0})+(1-\gamma) \varphi -g,\ldots,\gamma u_{k}(tt_{0}as_{1}s_{0})+(1- \gamma)\varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\gamma\sup_{t\in G} \bigl\Vert \bigl(u_{1}(tt_{0}as_{1}s_{0})-J_{1}(tt_{0})u_{1}(as_{1}s_{0}), \ldots,u_{k}(tt_{0}as_{1}s_{0})-J_{k}(tt_{0})u_{k}(as_{1}s_{0}) \bigr) \bigr\Vert _{k} \\& \qquad {}+\sup_{t\in G} \bigl\Vert \bigl(\gamma J_{1}(tt_{0})u_{1}(as_{1}s_{0}) + (1-\gamma)\varphi -g,\ldots,\gamma J_{k}(tt_{0})u_{k}(as_{1}s_{0}) + (1-\gamma)\varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\varphi(as_{1}s_{0}) + \sup_{t\in G} \bigl\Vert \bigl(\gamma J_{1}(tt_{0})u_{1}(as_{1}s_{0}) + (1-\gamma)\varphi- J_{1}(tt_{0}) \bigl(\gamma u_{1}(as_{1}s_{0}) + (1-\gamma)\varphi\bigr), \ldots, \\& \qquad \gamma J_{k}(tt_{0})u_{k}(as_{1}s_{0}) + (1-\gamma)\varphi- J_{k}(tt_{0}) \bigl(\gamma u_{k}(as_{1}s_{0}) + (1-\gamma)\varphi\bigr)\bigr) \bigr\Vert _{k} \\& \qquad {}+ \sup_{t\in G} \bigl\Vert \bigl(J_{1}(tt_{0}) \bigl(\gamma u_{1}(as_{1}s_{0}) + (1-\gamma ) \varphi\bigr)-g,\ldots, \\& \qquad J_{k}(tt_{0}) \bigl(\gamma u_{k}(as_{1}s_{0}) + (1-\gamma)\varphi\bigr)-g \bigr) \bigr\Vert _{k} \\& \qquad {}+\epsilon\sup_{t\in G}\bigl(1+\alpha(tt_{0}) \bigr)\xi^{-1} \bigl( \bigl\Vert \bigl(u_{1}(as_{1}s_{0})- \varphi,\ldots,u_{k}(as_{1}s_{0})-\varphi\bigr) \bigr\Vert _{k} \\& \qquad {}- \bigl\Vert \bigl(J_{1}(tt_{0})u_{1}(as_{1}s_{0})- \varphi,\ldots,J_{k}(tt_{0})u_{k}(as_{1}s_{0})- \varphi \bigr) \bigr\Vert _{k} + d\cdot\alpha(tt_{0}) \bigr) \\& \qquad {}+ \sup_{t\in G}\bigl(1+\alpha(tt_{0})\bigr) \bigl\Vert \bigl(\gamma u_{1}(as_{1}s_{0}) + (1- \gamma ) \varphi-g,\ldots,\gamma u_{k}(as_{1}s_{0}) + (1-\gamma) \varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\epsilon+ (1-\epsilon)\sup_{t\in G}\xi^{-1} \bigl( \bigl\Vert \bigl(u_{1}(as_{1}s_{0}) - \varphi,\ldots,u_{k}(as_{1}s_{0}) - \varphi\bigr) \bigr\Vert _{k} \\& \qquad {}- \bigl\Vert \bigl(u_{1}(tt_{0}as_{1}s_{0})- \varphi,\ldots,u_{k}(tt_{0}as_{1}s_{0})- \varphi\bigr) \bigr\Vert _{k}+ \varphi(as_{1}s_{0}) + \epsilon \bigr) \\& \qquad {}+ (1+ \epsilon) \bigl\Vert \bigl(\gamma u_{1}(as_{1}s_{0}) + (1-\gamma) \varphi-g,\ldots,\gamma u_{k}(as_{1}s_{0}) + (1-\gamma ) \varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\epsilon+ (1+\epsilon)\xi^{-1}(5\epsilon) \\& \qquad {}+(1+\epsilon) \bigl\Vert \bigl(\gamma u_{1}(as_{1}s_{0}) + (1- \gamma) \varphi-g,\ldots,\gamma u_{k}(as_{1}s_{0}) + (1-\gamma) \varphi-g\bigr) \bigr\Vert _{k}. \end{aligned}$$
Then
$$\begin{aligned}& \inf_{s\in G}\sup_{t\in G} \bigl\Vert \bigl( \gamma u_{1}(ts) + (1-\gamma)\varphi -g,\ldots,\gamma u_{k}(ts) + (1-\gamma)\varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\epsilon(1+\epsilon) \xi^{-1}(5\epsilon) \\& \qquad {}+(1+\epsilon)\inf _{a\in G} \bigl\Vert \bigl(\gamma u_{1}(as_{1}s_{0}) + (1-\gamma) \varphi-g,\ldots,\gamma u_{k}(as_{1}s_{0}) + (1-\gamma) \varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\epsilon(1+\epsilon) \xi^{-1}(5\epsilon) \\& \qquad {}+(1+\epsilon)\sup _{b\in G}\inf_{a\in G} \bigl\Vert \bigl(\gamma u_{1}(ab) + (1-\gamma) \varphi -g,\ldots,\gamma u_{k}(ab) + (1-\gamma) \varphi-g\bigr) \bigr\Vert _{k}. \end{aligned}$$
Hence,
$$\begin{aligned}& \inf_{s\in G}\sup_{t\in G} \bigl\Vert \bigl( \gamma u_{1}(ts) + (1-\gamma)\varphi -g,\ldots,\gamma u_{k}(ts) + (1-\gamma)\varphi-g\bigr) \bigr\Vert _{k} \\& \quad \leq\sup_{s\in G} \inf_{t\in G} \bigl\Vert \bigl(\gamma u_{1}(ts) + (1-\gamma )\varphi-g,\ldots,\gamma u_{k}(ts) + (1-\gamma)\varphi-g\bigr) \bigr\Vert _{k} \end{aligned}$$
because \(\epsilon>0\) is arbitrary. □

Theorem 2.5

Assume that \(\{(X^{k},\|\cdot\|_{k})\}_{k\in\mathbb{N}} \) is a uniformly convex multi-Banach space, and suppose that \(\emptyset\neq C\subset X\) is bounded and closed. Assume that \(\Im_{k} = \{J_{k}(t): t\in G\}\) for each \(k\geq1\) is a reversible semigroup of asymptotically nonexpansive functions on C. If D has a left invariant mean, then there exists a retraction \(P_{k}\) from \(\operatorname{LAO}(\Im_{k})\) onto \(F(\Im_{k})\) in which:
  1. (1)
    \(P_{k}\) is nonexpansive in the sense
    $$\begin{aligned}& \bigl\Vert (P_{1}u_{1}-P_{1}v_{1}, \ldots,P_{k}u_{k}-P_{k}v_{k}) \bigr\Vert _{k} \\& \quad \leq\inf_{s\in G}\sup_{t\in G} \bigl\Vert \bigl(u_{1}(st)-v_{1}(st),\ldots,u_{k}(st)-v_{k}(st) \bigr) \bigr\Vert _{k},\quad \forall u_{k},v_{k} \in \operatorname{LAO}(\Im_{k}); \end{aligned}$$
     
  2. (2)

    \(P_{k}J_{k}(p)u_{k} = J_{k}(p)P_{k}u_{k} = P_{k}u_{k} \) for all \(u_{k} \in \operatorname{AO}(\Im _{k})\) and \(p\in G\);

     
  3. (3)

    \(P_{k}u_{k} \in\bigcap_{s\in G}\overline{\operatorname{conv}}\{u_{k}(t): t\geq s\} \) for all \(u_{k}\in \operatorname{LAO}(\Im_{k})\).

     

Proof

We know D has a left invariant mean, so there is a net \(\{\gamma _{k,\alpha} : \alpha\in A\}\) of finite means on G in which \(\lim_{\alpha\in A} \|(\gamma_{1,\alpha}- \ell_{s}^{*}\gamma_{1,\alpha },\ldots,\gamma_{k,\alpha}- \ell_{s}^{*}\gamma_{k,\alpha})\|_{k}=0\) for every \(s\in G\), in which A is a directed system. Putting \(I = A \times G = \{\beta= (\alpha,t) : \alpha\in A, t\in G\}\). For \(\beta_{i} = (\alpha _{i},t_{i})\in I \), \(i =1,2\), define \(\beta_{1} \leq\beta_{2}\) iff \(\alpha_{1}\leq \alpha_{2} \), \(t_{1}\leq t_{2}\). Then, I is also a directed system. For each \(\beta=(\alpha,t)\in I\), define \(P_{k,1}\beta=\alpha\), \(P_{k,2}\beta=t\), and \(\gamma_{\beta}=\gamma _{\alpha}\). So, for every \(s\in G\),
$$ \lim_{\beta\in I} \bigl\Vert \bigl( \gamma_{1,\beta}-\ell^{*}\gamma_{1,\beta},\ldots,\gamma _{k,\beta}- \ell^{*}\gamma_{k,\beta}\bigr) \bigr\Vert _{k}=0. $$
(2.1)
Assume that \(\gamma= \{\{t_{\beta}\}_{\beta\in I}, t_{\beta} \geq P_{k,2}\beta,\forall\beta\in I\}\). Taking any \(\{t_{\beta},\beta\in I\} \in\gamma\), since \(r_{t\beta}^{*}\gamma_{k,\beta}\) is bounded, without loss of generality, let \(r_{t\beta}^{*}\gamma_{k,\beta}\) be \(weak^{*}\) convergent. Then, for all \(u_{k} \in \operatorname{LAO}(\Im_{k})\), \(\omega\mbox{-}\lim_{\beta\in I}\gamma_{k,\beta }(t)\langle u_{k}(tt_{\beta})\rangle\) exist. We define
$$P_{k}u_{k}=\omega\mbox{-}\lim_{\beta\in I} \gamma_{k,\beta}(t)\bigl\langle u_{k}(tt_{\beta})\bigr\rangle . $$
On the other hand, for every \(u_{k}\in \operatorname{LAO}(\Im_{k})\), \(P_{k}u_{k}\in\bigcap_{s\in G}\overline{\operatorname{conv}}\{u(t) : t \geq s\}\). Next, we shall show that \(P_{k}u_{k}\in F(\Im_{k})\). Then, for every \(\epsilon\in(0,1]\), there is \(t_{0} \in G\) such that, for each \(t\geq t_{0}\), \(\varphi(t)<\frac {a(\epsilon)}{4}\). Also, we can suppose that \(P_{k2}\beta\geq t_{0}\) for every \(\beta\in I\), so \(t_{\beta}\geq t_{0}\), \(\{t_{\beta}\}\in\gamma\). From Lemma 2.3, for every \(p\in G_{a(a(\frac{a(\epsilon )}{16}))}\), there is \(n\in N\) such that, for each \(t\in G\) and \(\beta \in I\),
$$\frac{1}{n}\sum_{i=1}^{n}J_{k} \bigl(p^{i}\bigr)u_{k}(tt_{\beta})\in F_{\frac{a(\epsilon )}{4}} \bigl(J_{k}(p)\bigr). $$
Since for every \(t\in G\)
$$\begin{aligned}& \Biggl\Vert \Biggl(\frac{1}{n}\sum_{i=1}^{n}J_{1} \bigl(p^{i}\bigr)u_{1}(tt_{\beta})- \frac{1}{n} \sum_{i=1}^{n}u_{1} \bigl(p^{i}tt_{\beta}\bigr),\ldots,\frac{1}{n}\sum _{i=1}^{n}J_{k}\bigl(p^{i} \bigr)u_{k}(tt_{\beta})- \frac{1}{n}\sum _{i=1}^{n}u_{k}\bigl(p^{i}tt_{\beta } \bigr)\Biggr) \Biggr\Vert _{k} \\& \quad \leq\varphi(tt_{\beta})< \frac{a(\epsilon)}{4}, \end{aligned}$$
we have, for every \(p\in G_{a(a(\frac{a(\epsilon)}{16}))}\),
$$\frac{1}{n}\sum_{i=1}^{n}u_{k} \bigl(p^{i}tt_{\beta}\bigr)\in F_{\frac{a(\epsilon )}{4}} \bigl(J_{k}(p)\bigr)+B\biggl(0,\frac{a(\epsilon)}{4}\biggr)\subset F_{a(\epsilon)}\bigl(J_{k}(p)\bigr). $$
Equation (2.1) implies that
$$\begin{aligned}& \lim_{{\beta}\in I} \Biggl\Vert \Biggl(\gamma_{1,{\beta}}(t)\Biggl\langle \frac{1}{n}\sum_{i=1}^{n}u_{1} \bigl(p^{i}tt_{\beta}\bigr)\Biggr\rangle -\gamma_{1,{\beta}} \bigl\langle u_{1}(tt_{\beta})\bigr\rangle ,\ldots, \\& \quad \gamma_{k,{\beta}}(t)\Biggl\langle \frac{1}{n}\sum _{i=1}^{n}u_{k}\bigl(p^{i}tt_{\beta} \bigr) \Biggr\rangle -\gamma_{k,{\beta}}\bigl\langle u_{k}(tt_{\beta}) \bigr\rangle \Biggr) \Biggr\Vert _{k}=0. \end{aligned}$$
Combining it with the definition of \(P_{k}u_{k}\), we get, for all \(p\in G_{a(a(\frac{a(\epsilon)}{16}))}\),
$$P_{k}u_{k}=\omega\mbox{-}\lim_{{\beta}\in I} \gamma_{k,{\beta}}(t)\Biggl\langle \frac {1}{n}\sum _{i=1}^{n}u_{k}\bigl(p^{i}tt_{\beta} \bigr)\Biggr\rangle \in\overline {\operatorname{co}}\, F_{a(\epsilon)} \bigl(J_{k}(p)\bigr). $$
Lemma 2.1 also implies that for every \(p\in G_{a(a(\frac {a(\epsilon)}{16}))}\), \(P_{k}u_{k}\in F_{\epsilon}(J_{k}(p))\). Now, the continuity of \(J_{k}(p)\) implies that \(P_{k}u_{k}\in F(\Im_{k})\). Obviously, for any \(p\in G\),
$$\begin{aligned} P_{k}J_{k}(p)u_{k} =& \omega\mbox{-}\lim _{{\beta}\in I}\gamma_{k,{\beta }}(t)\bigl\langle J_{k}(p)u_{k}(tt_{\beta}) \bigr\rangle \\ = & \omega\mbox{-}\lim_{{\beta}\in I}\gamma_{k,{\beta}}(t)\bigl\langle u_{k}(htt_{\beta })\bigr\rangle \\ = & \omega\mbox{-}\lim_{{\beta}\in I}\gamma_{k,{\beta}}(t)\bigl\langle u_{k}(tt_{\beta})\bigr\rangle \quad (\mbox{using (2.1)}) \\ = & P_{k}u_{k} \end{aligned}$$
and for every \(v_{k}\in \operatorname{LAO}(\Im_{k})\) and \(s\in G\), we have
$$\begin{aligned}& \bigl\Vert (P_{1}u_{1} - P_{1}v_{1}, \ldots,P_{1}u_{1} - P_{1}v_{1}) \bigr\Vert _{k} \\& \quad \leq\liminf_{{\beta}\in I} \bigl\Vert \bigl( \gamma_{1,{\beta}}(t)\bigl\langle u_{1}(tt_{\beta })\bigr\rangle - \gamma_{1,{\beta}}(t)\bigl\langle v_{1}(tt_{\beta}) \bigr\rangle ,\ldots,\gamma_{k,{\beta }}(t)\bigl\langle u_{k}(tt_{\beta}) \bigr\rangle - \gamma_{k,{\beta}}(t)\bigl\langle v_{k}(tt_{\beta}) \bigr\rangle \bigr) \bigr\Vert _{k} \\& \quad = \liminf_{{\beta}\in I} \bigl\Vert \bigl(\gamma_{1,{\beta}}(t) \bigl\langle u_{1}(stt_{\beta})\bigr\rangle - \gamma_{1,{\beta}}(t)\bigl\langle v_{1}(stt_{\beta})\bigr\rangle ,\ldots, \\& \qquad \gamma_{k,{\beta}}(t)\bigl\langle u_{k}(stt_{\beta}) \bigr\rangle - \gamma_{k,{\beta }}(t)\bigl\langle v_{k}(stt_{\beta}) \bigr\rangle \bigr) \bigr\Vert _{k} \quad (\mbox{by (2.1)}) \\& \quad \leq\liminf_{{\beta}\in I} \bigl\Vert \bigl( \gamma_{1,{\beta}}(t),\ldots,\gamma _{1,{\beta}}(t)\bigr) \bigr\Vert _{k} \cdot \sup_{t\in G} \bigl\Vert \bigl(u_{1}(stt_{\beta})-v_{1}(stt_{\beta }), \ldots,u_{k}(stt_{\beta})-v_{k}(stt_{\beta}) \bigr) \bigr\Vert _{k} \\& \quad \leq\sup_{t\in G} \bigl\Vert \bigl(u_{1}(st)-v_{1}(st), \ldots,u_{k}(st)-v_{k}(st)\bigr) \bigr\Vert _{k}. \end{aligned}$$
Thus,
$$\bigl\Vert (P_{1}u_{1} - P_{1}v_{1}, \ldots,P_{k}u_{k} - P_{k}v_{k}) \bigr\Vert _{k} \leq\inf_{s\in G}\sup_{t\in G} \bigl\Vert \bigl(u_{1}(st)-v_{1}(st),\ldots,u_{k}(st)-v_{k}(st) \bigr) \bigr\Vert _{k}. $$
 □

Theorem 2.6

(Ergodic theorem [17])

Assume that \(\{(X^{k},\|\cdot\|_{k})\}_{k\in\mathbb{N}} \) is a uniformly convex multi-Banach space, and suppose that \(\emptyset\neq C\subset X\) is bounded and closed. Assume that \(\Im_{k} = \{J_{k}(t) : t\in G\}\) is a reversible semigroup of asymptotically nonexpansive functions on C. If D has a left invariant mean and there is a unique retraction \(P_{k}\) from \(\operatorname{LAO}(\Im_{K})\) onto \(F(\Im_{k})\), which satisfies properties (1)(3) in Theorem 2.5, then for every strongly net \(\{\nu_{k,{\alpha }} : \alpha\in A\}\) on D and \(u_{k}\in \operatorname{AO}(\Im_{k})\),
$$\omega\mbox{-}\lim_{\alpha\in A} \int u_{k}(tp)\, d\nu_{k,\alpha}(t) = P_{k}\in F( \Im _{k})\quad \textit{uniformly in }p\in\gamma(G), $$
in which \(\gamma(G)=\{s\in G : st=ts \textit{ for all } t\in G \}\).

Theorem 2.7

Assume that \(\{(X^{k},\|\cdot\|_{k})\}_{k\in\mathbb{N}} \) is a uniformly convex multi-Banach space, and suppose that \(\emptyset\neq C\subset X\) is bounded and closed. Assume that \(\Im_{k} = \{J_{k}(t) : t\in G\}\) of a reversible semigroup of asymptotically nonexpansive mappings on C, and let \(u_{k}(\cdot)\) be an almost orbit of \(\Im_{k}\). If
$$\omega\mbox{-}\lim_{t\in G}u_{k}(pt)-u_{k}(t)=0 $$
for every \(p\in G\), then
$$\omega_{\omega}(u_{k})\subset F(\Im_{k}). $$

Proof

Let \(\epsilon\in(0,1]\), then there is \(t_{0}\in G\) such that, for \(t\geq t_{0}\), \(\varphi(t)<\frac{a(\epsilon)}{4}\). Suppose that \(p_{k}\in \omega_{\omega}(u_{k})\), so we can find a subnet \(\{u_{k}(t_{\alpha})\} _{\alpha\in A}\) of \(\{u_{k}(t)\}_{t\in G}\) with \(\omega\mbox{-}\lim_{\alpha\in A}u_{k}(t_{\alpha})=p_{k}\) in which, for every \(\alpha\in A\), \(t_{\alpha} \geq t_{0}\), in which A is a directed system. Using Lemma 2.3, for every \(p\in G_{a(a(\frac{a(\epsilon)}{16}))}\), we can find \(n\in\mathbb{N}\) such that, for every \(\alpha\in A\),
$$\frac{1}{n}\sum_{i=1}^{n} J_{k}\bigl(p^{i}\bigr)u_{k}(t_{\alpha})\in F_{\frac{a(\epsilon )}{4}}\bigl(J_{k}(p)\bigr). $$
Since for each \(\alpha\in A\)
$$\begin{aligned}& \Biggl\Vert \Biggl(\frac{1}{n}\sum_{i=1}^{n} J_{1}\bigl(p^{i}\bigr)u_{1}(t_{\alpha})- \frac{1}{n}\sum_{i=1}^{n} u_{1}\bigl(p^{i}t_{\alpha}\bigr),\ldots, \frac{1}{n}\sum_{i=1}^{n} J_{k}\bigl(p^{i}\bigr)u_{k}(t_{\alpha})- \frac{1}{n}\sum_{i=1}^{n} u_{k}\bigl(p^{i}t_{\alpha}\bigr)\Biggr) \Biggr\Vert _{k} \\& \quad \leq\varphi(t_{\alpha})< \frac{a(\epsilon)}{4}, \end{aligned}$$
we get
$$\frac{1}{n}\sum_{i=1}^{n} u_{k}\bigl(p^{i}t_{\alpha}\bigr)\in\frac{1}{n} \sum_{i=1}^{n} u_{k} \bigl(p^{i}t_{\alpha}\bigr)+B\biggl(0,\frac{a(\epsilon)}{4}\biggr) \subset F_{a(\epsilon)}\bigl(J_{k}(p)\bigr). $$
Since \(u_{k}(pt)-u_{k}(t) \rightarrow0\) for every \(p\in G\), we have \(u_{k}(p^{i}t_{\alpha})\rightarrow p_{k}\), \(i=1,2,\ldots,n\). Then, for all \(p\in G_{a(a(\frac{a(\epsilon)}{16}))}\),
$$p_{k} = \omega\mbox{-}\lim_{\alpha\in A}\frac{1}{n} \sum_{i=1}^{n} u_{k} \bigl(p^{i}t_{\alpha }\bigr)\in\overline{\operatorname{co}}\, F_{a(\epsilon)} \bigl(J_{k}(p)\bigr). $$
So, Lemma 2.1, implies that for every \(p\in G_{a(a(\frac {a(\epsilon)}{16}))}\), \(p\in F_{\epsilon}(J(p))\), hence \(p_{k}\in F(\Im_{k})\). □

In three last theorems X has not a Frechet differentiable norm.

Theorem 2.8

Assume that \(\{(X^{k},\|\cdot\|_{k})\}_{k\in\mathbb{N}} \) is a uniformly convex multi-Banach space with the Kadec–Klee property for its dual, and \(\emptyset\neq C\subset X\) is bounded and closed. Suppose that \(\Im_{k} = \{J_{k}(t) : t\in G\}\) of a reversible semigroup of asymptotically nonexpansive function on C and \(u_{k}(\cdot)\) is an almost orbit of \(\Im _{k}\). Then the following statements are equivalent:
  1. (1)

    \(\omega_{\omega}(u_{k})\subset F(\Im_{k})\);

     
  2. (2)

    \(\omega\mbox{-}\lim_{t\in G}u_{k}(t) = p_{k}\in F(\Im_{k})\);

     
  3. (3)

    \(\omega\mbox{-}\lim_{t\in G}u_{k}(pt)-u_{k}(t) = 0\) for every \(p\in G\).

     

Proof

(1) (2). It is enough to prove that \(\omega_{\omega}(u_{k})\) is a singleton. The reflexivity of X implies that \(X\neq\emptyset\). Suppose that \(\varphi_{k}\) and \(g_{k}\) are two elements in \(\omega_{\omega }(u_{k})\), then by (1) we get \(\varphi,g \in F(\Im_{k})\). For every \(\gamma\in(0,1)\), using Lemma 2.4, we have \(\lim_{t\in G}\|(\gamma u_{1}(t) + (1-\gamma)\varphi-g,\ldots,\gamma u_{k}(t) + (1-\gamma)\varphi-g)\|_{k}\) exists. Put
$$h(\gamma)= \lim_{t\in G} \bigl\Vert \bigl(\gamma u_{1}(t) + (1-\gamma)\varphi -g,\ldots,\gamma u_{k}(t) + (1- \gamma)\varphi-g\bigr) \bigr\Vert _{k}, $$
then for given \(\epsilon>0\), there is \(t_{1}\in G\) such that, for every \(t>t_{1}\),
$$\bigl\Vert \bigl(\gamma u_{1}(t) + (1-\gamma)\varphi-g,\ldots,\gamma u_{k}(t) + (1-\gamma )\varphi-g\bigr) \bigr\Vert _{k}\leq h(\gamma)+\epsilon. $$
So, for every \(t\geq t_{1}\),
$$\bigl\langle \gamma u_{k}(t) + (1-\gamma)\varphi-g,j(\varphi-g) \bigr\rangle \leq\|\varphi -g \|\bigl(h(\gamma) + \epsilon\bigr), $$
in which \(j(\varphi-g)\in J(\varphi-g)\). Let us note \(\varphi\in \overline{\operatorname{co}}\{u_{k}(t) : t\geq t_{1} \}\), so
$$\bigl\langle \gamma\varphi+ (1-\gamma)\varphi-g,j(\varphi-g)\bigr\rangle \leq\| \varphi-g \|\bigl(h(\gamma)+\epsilon\bigr), $$
which means \(\|\varphi-g\|\leq h(\gamma)+\epsilon\). We know ϵ is arbitrary, then
$$\|\varphi-g\|\leq h(\gamma). $$
\(g\in\omega_{\omega}(u_{k})\) implies that there is a subnet \(\{ u_{k}(t-{\alpha})\}_{\alpha\in A}\) in \(\{u_{k}(t)\}_{t\in G}\) such that \(\omega\mbox{-}\lim_{\alpha\in A}u_{k}(t_{\alpha})=g\), in which A is a directed system. Setting
$$I=A\times\mathbb{N} = \bigl\{ \beta= (\alpha,n) : \alpha\in A, n\in\mathbb {N} \bigr\} , $$
then for \(\beta_{i} =(\alpha_{i},n_{i})\), \(i \in I\), \(i=1,2\), define \(\beta_{1}\leq \beta_{2}\) iff \(\alpha_{1}\leq\alpha_{2}\), \(n_{1}\leq n_{2}\). For arbitrary \(\beta = (\alpha,n)\in I\), define \(P_{k,1}\beta= \alpha\), \(P_{k,2}\beta= n\), \(t_{\beta}= t_{\alpha}\), \(\epsilon_{\beta}=\frac{1}{P_{k,2}\beta}\). Then \(\omega\mbox{-}\lim_{\beta\in I}u_{k}(t_{\beta})= g\) and \(\lim_{\beta\in I}\epsilon_{\beta}= 0\). Using Lemma 1.6 implies that
$$\begin{aligned}& \bigl\Vert \bigl(\gamma u_{1}(t_{\beta}) + (1-\gamma) \varphi-g,\ldots,\gamma u_{k}(t_{\beta }) + (1-\gamma)\varphi-g \bigr) \bigr\Vert _{k} \\& \quad \leq \Vert \varphi-g \Vert ^{2} + 2\gamma\bigl\langle u_{k}(t_{\beta})-\varphi ,j\bigl(\gamma u_{k}(t_{\beta})+ (1-\gamma)\varphi-g\bigr)\bigr\rangle . \end{aligned}$$
Using Lemma 2.4 and the inequality \(\|\varphi-g\|\leq h(\gamma )\) implies that
$$\liminf_{\beta\in I}\bigl\langle u_{k}(t_{\beta}- \varphi,j\bigl(\gamma u_{k}(t_{\beta })+(1-\gamma)\varphi-g\bigr) \bigr\rangle \geq0. $$
So, for each \(\xi\in I\), there is \(\beta_{\xi}\in I\) such that \(\beta _{\xi}\geq\gamma\) and
$$ \bigl\langle u_{k}(t_{\beta_{\xi}})-\varphi,j\bigl( \epsilon_{\xi}u_{k}(t_{\beta_{\xi }})+(1-\epsilon_{\xi} \varphi-g)\bigr)\bigr\rangle \geq-\epsilon_{\xi}. $$
(2.2)
It is well known that \(\{\beta_{\xi}\}\) is also a subnet of I, then \(\omega\mbox{-}\lim_{\xi\in I}u_{k}(t_{\beta_{\xi}})=g\). Set
$$j_{\xi} = j\bigl(\epsilon_{\xi}u_{k}(t_{\beta_{\xi}})+(1- \epsilon_{\xi}\varphi-g)\bigr). $$
The reflexivity of X implies that \(X^{*}\) is also reflexive, and therefore the set of all weak limit points of \(\{j_{\xi}, \xi\in I\}\) is nonempty. Then, without loss of generality, let \(\omega\mbox{-} \lim_{\xi\in I}j_{\xi }=j\in X^{*}\). Then \(\|j\|\leq\liminf_{\xi\in I}\|j_{\xi}\| = \|\varphi -g\|\). Since
$$\langle \varphi-g,j_{\xi}\rangle =\bigl\Vert \epsilon_{\xi}u_{k}\bigl(t_{\beta_{\xi}}+(1- \epsilon _{\xi})\varphi-g\bigr)\bigr\Vert ^{2}-\epsilon_{\xi}\bigl\langle u_{k}(t_{\beta_{\xi}}-\varphi,j_{\xi}\bigr\rangle . $$
Passing the limit for \(\xi\in I\), we get \(\langle \varphi-g,j\rangle =\|\varphi-g\| ^{2}\), which implies \(\|j\|\geq\|\varphi-g\|\). Then
$$\langle \varphi-g,j\rangle =\|\varphi-g\|^{2}=\|j\|^{2}, $$
i.e., \(j\in J(\varphi-g)\). Hence, \(\omega\mbox{-}\lim_{\xi\in I}j_{\xi}=j\) and \(\lim_{\xi\in I}\|j_{\xi}\|=\|j\|\). By the reflexivity of \(X^{*}\) and the Kadec–Klee property, we conclude that \(\lim_{\xi\in I}j_{\xi}=j\). Take the limit for \(\xi\in I\) in 2.2, we get \(\langle g-\varphi,j\rangle \geq0\), i.e., \(\|\varphi-g\|^{2}\leq0\), which implies \(\varphi=g\).

(2) (3). Obviously.

(3) (1). See Theorem 2.7. □

Declarations

Acknowledgements

We would like to thank the referee(s) for his comments and suggestions on the manuscript.

Funding

No funding was received.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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

(1)
Science and Research Branch, IAU, Tehran, Iran
(2)
Department of Mathematics, Iran University of Science and Technology, Tehran, Iran
(3)
Research Institute for Natural Sciences, Hanyang University, Seoul, Republic of Korea

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