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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

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.

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 [1,2,3].

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 [4,5,6,7,8,9]).

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 [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. 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. □

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Kenari, H.M., Saadati, R. & Park, C. 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. J Inequal Appl 2019, 43 (2019). https://doi.org/10.1186/s13660-019-1996-8

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