Abstract
In this paper, we consider random multi-normed spaces introduced by Dales and Polyakov (Multi-Normed Spaces, 2012). Next, by the fixed point method, we approximate the multiplicatives on these spaces.
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1 Introduction
The concept of random normed spaces and their properties are discussed in [2]. Also, the concept of multi-normed spaces was introduced by Dales and Polyakov. In this paper we combine the mentioned concepts and introduce random multi-normed spaces. Next, we get an approximation for homomorphisms in these spaces. For more results and applications, one can see [3–23].
Definition 1.1
Let \((E,\mu,\ast)\) be a random normed space. ∗ is a continuous t-norm. A multi-random norm on \(\{ E^{k},k\in\mathbb{N} \} \) is sequence \(\{N_{k}\} \) such that \(N_{k} \) is a random norm on \(E^{k}\) (\(k\in\mathbb{N}\)), \(\mu_{x}^{1}(t)= \mu_{x}(t)\) for each \(x\in E\) and \(t\in\mathbb{R}\) and the following axioms are satisfied for each \(k\in\mathbb{N}\) with \(k\geq2\):
-
(NF1)
\(\mu_{A_{\sigma}(x)}^{k}(t)=\mu_{x}^{k}(t)\), for each \(\sigma\in\sigma_{k}\), \(x\in E^{k}\), \(t\in\mathbb{R}\),
-
(NF2)
\(\mu_{M_{\alpha}(x)}^{k}(t)\geq \mu_{\max_{i\in\mathbb{N}_{k}} \vert \alpha_{i} \vert x}^{k}(t)\), for each \(\alpha= (\alpha_{1},\ldots,\alpha_{k})\in\mathbb{R}^{k}\), \(x\in E^{k}\), \(t\in\mathbb{R}\),
-
(NF3)
\(\mu_{(x_{1},\ldots,x_{k},0)}^{k+1}(t)=\mu_{(x_{1},\ldots,x_{k})} ^{k}(t)\), for each \(x_{1},\ldots,x_{k}\in E\) and \(t\in\mathbb{R}\),
-
(NF4)
\(\mu_{(x_{1},\ldots,x_{k},x_{k})}^{k+1}(t)=\mu_{(x_{1},\ldots,x _{k})}^{k}(t)\), for each \(x_{1},\ldots,x_{k}\in E\) and \(t\in\mathbb{R}\).
In this case \(\{(E^{k},\mu^{k},\ast),k\in\mathbb{N} \}\) is called a random multi-normed space. Moreover, if axiom (NF4) is replaced by the following axiom:
-
(DF4)
\(\mu_{(x_{1},\ldots,x_{k},x_{k})}^{k+1}(t)=\mu_{(x_{1},\ldots,2x _{k})}^{k}(t)\), for each \(x_{1},\ldots,x_{k}\in E\) and \(t\in\mathbb{R}\),
then \(\{\mu^{k} \}\) is called a dual random multi-normed and \(\{(E ^{k},\mu^{k},\ast), k\in\mathbb{N} \}\) is called a dual random multi-normed space.
2 Approximation of the multiplicatives
We apply fixed point theory [24] to get an approximation for multiplicatives. A metric d on non-empty set ϒ with range \([0,\infty]\) is called a generalized metric.
Lemma 2.1
Let \(k\in\mathbb{N}\), and let E and F be linear spaces such that \((F^{k},\mu^{k},*)\) is a complete random multi-normed space. Let there exist \(0\leq M<1\), \(\lambda> 0\), and a function \(\psi: E^{k} \longrightarrow[0,\infty)\) such that
We set \(\Upsilon:= \{ \eta:E \longrightarrow F:\eta(0)=0 \}\), and define \(d:\Upsilon\times\Upsilon\) on \([0,\infty]\) by
Then \((\Upsilon,d)\) is a complete generalized metric space, and the mapping \(J: \Upsilon\longrightarrow\Upsilon\) defined by \((Jg)(x):=\frac{g( \lambda x)}{\lambda}\) (\(x\in\Upsilon\)) is a strictly contractive mapping.
Theorem 2.2
Let E be a linear space and let \(((F^{n},\mu ^{n},*):n\in\mathbb{N})\) be a complete random multi-normed space. Let \(k\in\mathbb{N}\) and let there exist \(0\leq M_{0} < 1\) and a function \(\varphi:E^{2k} \longrightarrow [0,\infty)\) satisfying
for all \(x_{1},y_{1},\ldots,x_{k},y_{k}\in E\). Suppose that \(f:E \longrightarrow F\) is a mapping with \(f(0)=0\) and
for all \(\lambda\in\mathbb{T}:=\{\lambda\in\mathbb{C}: \vert \lambda \vert =1\}\) and \(x_{1},y_{1},\ldots,x_{k},y_{k}\in E\), \(t> 0\).
Then
exists for any \(x_{1},\ldots,x_{k} \in E\) and defines a random homomorphism \(H:E \longrightarrow F\) such that
for all \(x_{1},\ldots,x_{k} \in E\) and \(t> 0\).
Proof
Let \(x_{1}=\frac{x_{1}}{2},\ldots,x_{k}=\frac{x_{k}}{2}\), \(y_{1}=\frac{y _{1}}{2},\ldots,y_{k}=\frac{y_{k}}{2}\) in (2.2). We get
since f is odd, \(f(0)=0\). So \(\mu_{f(0)}(\frac{t}{2})=1\). Letting \(\lambda=1 \) and \(y= x\), we get
for all \(x_{1},y_{1},\ldots,x_{k},y_{k}\in E\). Consider the following set:
and introduce the generalized metric on s:
where, as usual, \(\inf\phi=+\infty\). It is easy to show \((s,d)\) is complete. Now, we consider the linear mapping \(J:s \longrightarrow s\) such that
for all \(x \in E\). Let \(g,h \in s\) be given such that \(d(g,h)=\varepsilon \). Then we have
for all \(x_{1},\ldots,x_{k} \in E\) and all \(t> 0\) and hence we have
for all \(x_{1},\ldots,x_{k} \in E\) and \(t> 0\). Then \(d(g,h)=\varepsilon \) implies that \(d(J{g},J{h})\leq M_{0}\varepsilon\). This means that
for all \(g,h\in s\). It follows that
for all \(x_{1},\ldots,x_{k} \in E\) and \(t> 0\). So \(d(f,J{f})\leq\frac{M _{0}}{2}\).
Now, there exists a mapping \(H: E \longrightarrow F\) satisfying the following:
-
(1)
H is a fixed point of J, i.e.,
$$ H \biggl( \frac{x}{2} \biggr) =\frac{1}{2}H(x) $$(2.10)for all \(x \in E\). Since \(f:E \longrightarrow E\) is odd, \(H:E \longrightarrow F\) is an odd mapping. The mapping H is a unique fixed point of J in the set
$$ M=\bigl\{ g\in s:d(f,g)< \infty\bigr\} . $$This implies that H is a unique mapping satisfying (2.10) such that there exists a \(\nu\in(0,\infty)\) satisfying
$$ \mu^{k}_{(f(x_{1})-H(x_{1}),\ldots,f(x_{k})-H(x_{k}))}(\nu t)\geq\frac{t}{t+ \varphi(x_{1},\ldots,x_{k})} $$for all \(x_{1},\ldots,x_{k} \in E\),
-
(2)
\(d(J^{n}f,H)\rightarrow0 \) as \(n\rightarrow\infty\). This implies that
$$ \lim_{n\rightarrow\infty}2^{n}f \biggl( \frac{x}{2_{n}} \biggr) =H(x) $$for all \(x \in E\),
-
(3)
\(d(f,H)\leq\frac{1}{1-M_{0}}d(f,Jf)\), which implies
$$ d(f,H)\leq\frac{M_{0}}{2-2M_{0}}. $$
Put \(\lambda=1\) in (2.3). Then
for all \(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k} \in E\), \(t> 0\) and \(n\geq1\). Since
for all \(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k} \in E\), \(t> 0\). It follows that
for all \(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k} \in E\), \(t> 0\). So mapping \(H:E \longrightarrow F\) is Cauchy additive.
Let \(y_{1}=x_{1},\ldots,y_{k}=x_{k}\) in (2.3). Then we have
for all \(\lambda,\beta\in\mathbb{T}\), \(\lambda=\frac{\beta}{2}\), \(x_{1},\ldots,x_{k} \in E\), \(t> 0\) and \(n\geq1\). So we have
for all \(\beta\in\mathbb{T}\), \(x_{1},\ldots,x_{k} \in E\), \(t> 0\) and \(n\geq1\). We have
for all \(x_{1},\ldots,x_{k} \in E\), \(t> 0\), and
for all \(\beta\in\mathbb{T}\), \(x_{1},\ldots,x_{k} \in E\), \(t> 0\). Thus, the additive mapping \(H:E \longrightarrow F\) is \(\mathbb{R}\)-linear. From (2.4), we have
for all \(x_{1},\ldots,x_{k} \in E\), \(t> 0\) and \(n\geq1\).
Then we have
for all \(x_{1},\ldots,x_{k} \in E\), \(t> 0\) and \(n\geq1\).
Since
for all \(x_{1},\ldots,x_{k} \in E\), \(t> 0\), we have
for all \(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k} \in E\), \(t> 0\). Thus, the mapping \(H:E \longrightarrow F\) is multiplicative. Therefore, there exists a unique random homomorphism \(H:E\longrightarrow F\) satisfying (2.6), and this completes the proof. □
3 Approximation in dual random multi-normed space
The following lemma is an immediate result of the definition of random multi-normed space.
Lemma 3.1
Let \(\{(E^{k},\mu^{k},\ast),k\in\mathbb{N}\}\) be a dual random multi-normed space, \(k,n\in\mathbb{N}\), \(x_{1},x_{2},\ldots ,x_{k},x_{k+1},\ldots,x _{k+n}\in\mathbb{E}\) and \(\lambda_{1},\ldots,\lambda_{k}\) be real numbers of absolute value 1. Then we have:
-
(i)
\(\mu_{(\lambda_{1}x_{1},\ldots,\lambda_{k}x_{k})}^{k}(t)= \mu_{(x_{1},\ldots,x_{k})}^{k}(t)\),
-
(ii)
\(\mu_{(x_{1},\ldots,x_{k})}^{k}(t)\geq \mu_{(x_{1},\ldots,x_{k},x_{k+1})}^{k+1}(t)\),
-
(iii)
\(\mu_{(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{k+n})}^{k+n}(t) \geq T_{M}(\mu_{(x_{1},\ldots,x_{k})}^{k}(\alpha t), \mu_{(x_{k+1},\ldots,x_{k+n})}^{n}(\beta t))\), where \(\alpha,\beta \geq0\) and \(\alpha+ \beta=1\),
-
(iv)
\(\min_{i\in{\mathbb{N}_{k}}} \mu_{x_{i}}(t)\geq \mu_{(x_{1},\ldots,x_{k})}^{k}(t)\geq\min_{i\in\mathbb{N}_{k}} \mu_{x _{i}}(\alpha_{i}t)\),
where \(\alpha_{1},\ldots,\alpha_{k} \geq0 \) and \(\sum_{i=1}^{k}\alpha _{i}=1\). In particular, we have
Theorem 3.2
Let E be a linear space, and \(\{(E^{k},\mu^{k},\ast),k\in \mathbb{N}\}\) be a random multi space. Let \(\alpha\in(0,1)\) and \(f:E\longrightarrow F\) is a mapping satisfying \(f(0)=0\) and
where \(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k}\in E\) and \(t,s\in\mathbb{N}\) with the greatest common divisor \((t,s)=1\).
Then there exists a unique additive mapping \(T:E\longrightarrow F\) such that
for all \(x_{1},\ldots,x_{k}\in E\) and \(t,s\in\mathbb{N}\) with \((t,s)=1\).
Proof
Replacing \(x_{1},\ldots,x_{k}\) and \(y_{1},\ldots,y_{k}\) by \(2x_{1},\ldots,2x _{k}\) and \(0,\ldots,0\) in (3.1), respectively, yields
Replacing \(x_{1},\ldots,x_{k}\), t, s by \(2x_{1},\ldots,2x_{k}\), 2t, 2s, respectively, in (3.3) and repeating this process for n-time (\(n\in\mathbb{N}\)), it follows that
for \(n,m\in\mathbb{N}\) with \(n> m\). Using (3.4) and (RN2) we get
Then
for \(x\in E\). Then, replacing \(x_{1},\ldots,x_{k}\) by \(x,2x,\ldots,2^{k-1}x\) in (3.5), we have
Let \(\varepsilon> 0\) be given. Then there exists \(n_{0} \in \mathbb{N}\) such that \(\frac{\alpha}{2^{n_{0}}}< \varepsilon\). Now we substitute m, n with n, \(n+p\) (\(p\in\mathbb{N}\)), respectively, in (3.6), for each \(n\geq n_{0}\), and we get
By Lemma 3.1, we have
for all \(n> n_{0}\) and \(p\in\mathbb{N}\). The density of rational numbers in \(\mathbb{R}\) is useful in checking correctness of (3.6) with positive real number r instead of \(\frac{t}{s}\). Then we have
for each \(x\in E\), \(r\in\mathbb{R^{+}}\), \(n\geq n_{0}\) and \(p\in\mathbb{N}\). Then \(\{\frac{f(2^{n}x)}{2^{n}}\}\) is a Cauchy sequence, so it is convergent in the random multi-Banach space \(\{(E^{k},\mu^{k},\ast),k\in\mathbb{N}\}\). Setting \(T(x):= \lim_{n\rightarrow\infty}\frac{f(2^{n}x)}{2^{n}}\) and applying again Lemma 3.1, for each \(r> 0\), we have
and
We put \(m= 0\) in (3.5), and we get
Then
by (3.8) and when \(n\rightarrow\infty\), which implies that (3.2).
Now, we show that T is additive. Let \(x,y\in E\) and replace \(x_{1},\ldots,x_{k}\) by \(2^{n}x\), \(y_{1},\ldots,y_{k}\) by \(2^{n}y\), and t by \(2^{n}t\) in (3.1). We get
Using (NF4), we conclude that
On the other hand, we obtain that
for each \(x,y\in E\), \(t,s\in\mathbb{N}\) with \((t,s)=1\). Utilizing again the density of \(\mathbb{Q}\) in \(\mathbb{R}\), we find that (3.11) remains true if \(\frac{4t}{s}\) is substituted with a positive real number r.
Consequently,
for each \(x,y\in E\) and \(r\in\mathbb{R}\). Letting \(n\rightarrow \infty\) reveals that T complies with Jensen, and using the fact that \(T(0)=0\), we conclude that T is additive [27, Theorem 6].
It remains to show the uniqueness of T. Suppose that \(T'\) is another additive mapping satisfying (3.2). Then, for each \(t,s\in \mathbb{N}\), sufficiently large n in \(\mathbb{N}\) and \(x\in E\),
This inequality holds for each \(r\in\mathbb{R}^{+}\) instead of \(\frac{t}{s}\), too. Therefore, for each \(r\in\mathbb{R}^{+}\), \(n\in\mathbb{N}\), \(\mu_{T'(x)-T(x)}(r)\geq1-\frac{\alpha}{2^{n-2}}\), letting \(n\rightarrow\infty\), it follows that \(T= T'\). □
4 Conclusion
In this paper, we consider multi-Banach spaces, approximate by multiplicatives, and provide some controlled mappings, which are stable by control functions.
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Agarwal, R.P., Saadati, R. & Salamati, A. Approximation of the multiplicatives on random multi-normed space. J Inequal Appl 2017, 204 (2017). https://doi.org/10.1186/s13660-017-1478-9
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DOI: https://doi.org/10.1186/s13660-017-1478-9