- Research
- Open access
- Published:
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 Applications volume 2019, Article number: 43 (2019)
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 ℓ-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 ℓ symbols.
Definition 1.1
Suppose that Y is a vector space, and take \(\ell\in\mathbb{N}\). For \(\sigma\in\varSigma_{\ell}\), define
For \(\beta= (\beta_{j}) \in{\mathbb {C}} ^{\ell}\), define
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 \(k\geq2\):
-
(a1)
for each \(\sigma\in\varSigma_{\ell}\) and \(y\in Y^{\ell}\), we have
$$\bigl\Vert B_{\sigma}(y) \bigr\Vert _{\ell}= \Vert y \Vert _{\ell}; $$ -
(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
$$\bigl\Vert K_{\beta}(y) \bigr\Vert _{\ell}\leq\Bigl(\max _{j\in{\mathbb {N}}_{\ell}} \vert \beta_{j} \vert \Bigr) \Vert y \Vert _{\ell}; $$ -
(a3)
for each \(y_{1},\ldots,y_{\ell-1}\), we have
$$\bigl\Vert (y_{1},\ldots,y_{\ell-1},0) \bigr\Vert _{\ell}= \bigl\Vert (y_{1},\ldots,y_{\ell-1}) \bigr\Vert _{\ell-1}; $$ -
(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
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
-
(a)
\(\|(y,\ldots,y)\|_{\ell}=\|y\|\) (\(y\in Y\));
-
(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
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
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
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
Let \(y\in Y\). We say that the sequence \((y_{m})\) is multi-convergent to \(y\in{Y}\) and write
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
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
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
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
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 \(\sum_{i=1}^{n} a_{i} =1\), \(u_{1},\ldots,u_{n} \in C\),
in which \(d = 4\sup\{\|u\| : u \in C\}+1\).
For every \(\epsilon\in(0,1]\), define
and
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 \(p\in G_{a({\epsilon})}\),
Proof
Since \(F_{\epsilon}(J_{K}(p))\) is closed, we only need to prove that, for all \(p\in G_{a({\epsilon})}\),
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
□
Lemma 2.2
For every \(p\in G_{\frac{\epsilon}{4}}\),
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
□
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 \(u\in C\),
Proof
Let \(\epsilon\in(0,1]\) and \(m= \frac{2d+1}{a(\frac{\epsilon}{4})}\). There is \(n_{0}\in N\) satisfying
For any \(n\geq n_{0}\) and \(p\in G_{a(a(\frac{\epsilon}{4}))}\), we can take a number
For every \(i\in N\) and \(u\in C\), we put
By \(\alpha(p)\leq\frac{1}{8}\) and
we get
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
Hence
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
\(i=1,2,\ldots,n\). Now, there are at most N elements in \(\{i=1,2,\ldots,n\}\) such that \(\ell_{i} \neq0 \). Since
we can conclude that, for all \(p \in G_{a(a(\frac{\epsilon}{4}))}\),
By Lemma 2.1, we get, for all \(p \in G_{a(a(\frac{\epsilon }{4}))} \subset G_{a(\frac{\epsilon}{4})}\),
Using Lemma 2.2 and
we obtain
□
Lemma 2.4
Suppose that \(u_{k}(\cdot)\) is an almost orbit of \(\Im_{k}\). So
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
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 \(a\in G\),
Hence
Thus, there exists \(s_{1}\in G\) such that
Then, for every \(a\in G\), we get
Then
Hence,
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)
\(P_{k}\) 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)
\(P_{k}J_{k}(p)u_{k} = J_{k}(p)P_{k}u_{k} = P_{k}u_{k} \) for all \(u_{k} \in \operatorname{AO}(\Im _{k})\) and \(p\in G\);
-
(3)
\(P_{k}u_{k} \in\bigcap_{s\in G}\overline{\operatorname{conv}}\{u_{k}(t): t\geq s\} \) 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 \(s\in G\),
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
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 \(\beta \in I\),
Since for every \(t\in G\)
we have, for every \(p\in G_{a(a(\frac{a(\epsilon)}{16}))}\),
Equation (2.1) implies that
Combining it with the definition of \(P_{k}u_{k}\), we get, for all \(p\in G_{a(a(\frac{a(\epsilon)}{16}))}\),
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 \(p\in G\),
and for every \(v_{k}\in \operatorname{LAO}(\Im_{k})\) and \(s\in G\), we have
Thus,
□
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 \(u_{k}\in \operatorname{AO}(\Im_{k})\),
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
for every \(p\in G\), then
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 \(\alpha\in A\),
Since for each \(\alpha\in A\)
we get
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 \(p\in G_{a(a(\frac{a(\epsilon)}{16}))}\),
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)
\(\omega_{\omega}(u_{k})\subset F(\Im_{k})\);
-
(2)
\(\omega\mbox{-}\lim_{t\in G}u_{k}(t) = p_{k}\in F(\Im_{k})\);
-
(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
then for given \(\epsilon>0\), there is \(t_{1}\in G\) such that, for every \(t>t_{1}\),
So, for every \(t\geq t_{1}\),
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
which means \(\|\varphi-g\|\leq h(\gamma)+\epsilon\). We know ϵ is arbitrary, then
\(g\in\omega_{\omega}(u_{k})\) 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
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
Using Lemma 2.4 and the inequality \(\|\varphi-g\|\leq h(\gamma )\) implies that
So, for each \(\xi\in I\), there is \(\beta_{\xi}\in I\) such that \(\beta _{\xi}\geq\gamma\) and
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
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
Passing the limit for \(\xi\in I\), we get \(\langle \varphi-g,j\rangle =\|\varphi-g\| ^{2}\), which implies \(\|j\|\geq\|\varphi-g\|\). Then
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. □
References
Dales, H.G., Polyakov, M.E.: Diss. Math. 488, 165 pp. (2012)
Dales, H.G., Moslehian, M.S.: Glasg. Math. J. 49, 321–332 (2007)
Moslehian, M.S., Nikodem, K., Popa, D.: J. Math. Anal. Appl. 355, 717–724 (2009)
Day, M.M.: Ill. J. Math. 1, 509–544 (1957)
Holmes, R.D., Narayanaswamy, P.P.: Can. Math. Bull. 13, 209–214 (1970)
Holmes, R.D., Lau, A.T.: Bull. Lond. Math. Soc. 3, 343–347 (1971)
Krik, W., Torrejon, R.: Nonlinear Anal. 3, 111–121 (1979)
Saeidi, S.: J. Fixed Point Theory Appl. 5, 93–103 (2009)
Saeidi, S.: Fixed Point Theory Appl. 2009, Article ID 363257 (2009)
Chang, S.S.: Nonlinear Anal. 30, 4197–4208 (1997)
Bruck, R.E.: Isr. J. Math. 32, 107–116 (1979)
Baillon, J.B.: C. R. Math. Acad. Sci. Paris 280, A1511–A1514 (1976)
Takahashi, W.: Proc. Am. Math. Soc. 17, 55–58 (1986)
Li, G., Kim, J.K.: Nonlinear Anal. 55, 1–14 (2003)
Li, G., Kim, J.K.: Acta Math. Sci. 18, 25–30 (1998)
Lau, A.T., Shioji, N., Takahashi, W.: J. Funct. Anal. 161, 62–75 (1999)
Kim, J.K., Li, G.: Dyn. Syst. Appl. 9, 255–268 (2000)
Kaczor, W., Kuczumow, T., Michalska, M.: Nonlinear Anal. 67, 2122–2130 (2007)
Kim, K.S.: J. Math. Anal. Appl. 358, 261–272 (2009)
Li, G., Kim, J.K.: Houst. J. Math. 29, 23–36 (2003)
Li, G.: J. Math. Anal. Appl. 206, 451–464 (1997)
Ok, H.: Nonlinear Anal. 7, 619–635 (1992)
Saeidi, S.: Nonlinear Anal. 71, 2558–2563 (2009)
Saeidi, S.: Nonlinear Anal. 69, 3417–3422 (2008)
Falset, J.G., Kaczor, W., Kuczumow, T., et al.: Nonlinear Anal. 43, 377–401 (2001)
Kaczor, W.: J. Math. Anal. Appl. 272, 565–574 (2002)
Miyadera, I., Kobayasi, K.: Nonlinear Anal. 6, 349–356 (1982)
Aksoy, A.G., Khamsi, M.A.: Springer, New York (1990)
Kenari, H.M., Saadati, R., Cho, Y.J.: J. Inequal. Appl. 2014, Article ID 259 (2014)
Kenari, H.M., Saadati, R., Azhini, M. Cho, Y.J., J. Inequal. Appl. 2014, Article ID 402 (2014)
Zhu, L., Huang, Q., Li, G.: Fixed Point Theory Appl. 2013, Article ID 231 (2013)
Acknowledgements
We would like to thank the referee(s) for his comments and suggestions on the manuscript.
Funding
No funding was received.
Author information
Authors and Affiliations
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.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-019-1996-8