Open Access

Approximation of the multiplicatives on random multi-normed space

Journal of Inequalities and Applications20172017:204

Received: 15 May 2017

Accepted: 23 August 2017

Published: 31 August 2017


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.


approximation homomorphisms random multi-normed space dual random multi-normed space


39A10 39B52 39B72 46L05 47H10 46B03 54E40 54E35 54H25

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 [323].

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 \(k\geq2\):
  1. (NF1)

    \(\mu_{A_{\sigma}(x)}^{k}(t)=\mu_{x}^{k}(t)\), for each \(\sigma\in\sigma_{k}\), \(x\in E^{k}\), \(t\in\mathbb{R}\),

  2. (NF2)

    \(\mu_{M_{\alpha}(x)}^{k}(t)\geq \mu_{\max_{i\in\mathbb{N}_{k}} \vert \alpha_{i} \vert x}^{k}(t)\), for each \(\alpha= (\alpha_{1},\ldots,\alpha_{k})\in\mathbb{R}^{k}\), \(x\in E^{k}\), \(t\in\mathbb{R}\),

  3. (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 \(t\in\mathbb{R}\),

  4. (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:
  1. (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

[25, 26]

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
$$\begin{aligned} \psi(\lambda x_{1},\ldots,\lambda x_{k})\leq \lambda M\psi (x_{1},\ldots,x _{k}) \quad(x_{1}, \ldots,x_{k}\in E). \end{aligned}$$
We set \(\Upsilon:= \{ \eta:E \longrightarrow F:\eta(0)=0 \}\), and define \(d:\Upsilon\times\Upsilon\) on \([0,\infty]\) by
$$\begin{aligned}& d(\eta,\zeta) \\& \quad = \inf\biggl\{ c> 0: \mu_{(\eta(x_{1})-\zeta(x_{1}),\ldots,\eta(x_{k})-\zeta(x_{k}))}(ct) \geq\frac{t}{t+\psi(x_{1},\ldots,x_{k})}, x_{1},\ldots,x_{k} \in E \biggr\} . \end{aligned}$$

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
$$ \varphi(2x_{1},2y_{1},\ldots,2x_{k},2y_{k}) \leq2M_{0}\varphi(x_{1},y _{1}, \ldots,x_{k},y_{k}) $$
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
$$\begin{aligned}& \mu^{k}_{(f(\lambda x_{1}+\lambda y_{1})-\lambda f(x_{1})-\lambda f(y _{1}),\ldots,f(\lambda x_{k}+\lambda y_{k})-\lambda f(x_{k})-\lambda f(y _{k}))}(t) \\& \quad \geq \frac{t}{t+\varphi(x_{1},y_{1},\ldots,x_{k},y_{k})}, \end{aligned}$$
$$\begin{aligned}& \mu^{k}_{(f(x_{1}y_{1})-f(x_{1})f(y_{1}),\ldots ,f(x_{k}y_{k})-f(x_{k})f(y _{k}))}(t)\geq\frac{t}{t+\varphi(x_{1},y_{1},\ldots,x_{k},y_{k})}, \end{aligned}$$
for all \(\lambda\in\mathbb{T}:=\{\lambda\in\mathbb{C}: \vert \lambda \vert =1\}\) and \(x_{1},y_{1},\ldots,x_{k},y_{k}\in E\), \(t> 0\).
$$ H(x):=\lim_{n\rightarrow\infty}2^{n}f\biggl( \frac{x}{2^{n}}\biggr) $$
exists for any \(x_{1},\ldots,x_{k} \in E\) and defines a random homomorphism \(H:E \longrightarrow F\) such that
$$\begin{aligned}& \mu_{(f(x_{1})-H(x_{1}),\ldots,f(x_{k})-H(x_{k}))}(t) \geq \frac{(1-M _{0})t}{(1-M_{0})t+M_{0}\psi(x_{1},\ldots,x_{k})}, \end{aligned}$$
$$\begin{aligned}& \psi(x_{1},\ldots,x_{k})= \varphi\biggl( \frac{x_{1}}{2},\frac {x_{1}}{2},\ldots,\frac{x _{k}}{2},\frac{x_{k}}{2} \biggr), \end{aligned}$$
for all \(x_{1},\ldots,x_{k} \in E\) and \(t> 0\).


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
$$ \varphi(x_{1},y_{1},\ldots,x_{k},y_{k}) \leq2M_{0}\varphi\biggl( \frac{x_{1}}{2},\frac{y_{1}}{2},\ldots, \frac{x_{k}}{2},\frac{y_{k}}{2}\biggr), $$
since f is odd, \(f(0)=0\). So \(\mu_{f(0)}(\frac{t}{2})=1\). Letting \(\lambda=1 \) and \(y= x\), we get
$$ \mu^{k}_{(f(2x_{1})-2f(x_{1}),\ldots,f(2x_{k})-2f(x_{k}))}(t) \geq\frac{t}{t+ \varphi(x_{1},x_{1},\ldots,x_{k},x_{k})} $$
for all \(x_{1},y_{1},\ldots,x_{k},y_{k}\in E\). Consider the following set:
$$ s=:\{g:E \longrightarrow F \} $$
and introduce the generalized metric on s:
$$\begin{aligned}& d(g,h) \\& \quad= \inf\biggl\{ \nu\in\mathbb{R}_{+} :\mu^{k}_{(g(x_{1})-h(x_{1}),\ldots,g(x _{k})-h(x_{k}))}( \nu t)\geq\frac{t}{t+\varphi(x_{1},\ldots,x_{k})}, x _{1},\ldots,x_{k}\in E,t> 0 \biggr\} , \end{aligned}$$
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
$$ J\bigl(g(x)\bigr) :=2g \biggl( \frac{x}{2} \biggr) $$
for all \(x \in E\). Let \(g,h \in s\) be given such that \(d(g,h)=\varepsilon \). Then we have
$$ \mu^{k}_{(g(x_{1})-h(x_{1}),\ldots,g(x_{k})-h(x_{k}))}(\varepsilon t) \geq\frac{t}{t+\varphi(x_{1},x_{1},\ldots,x_{k},x_{k})}, $$
for all \(x_{1},\ldots,x_{k} \in E\) and all \(t> 0\) and hence we have
$$\begin{aligned} \mu^{k}_{(Jg(x_{1})-Jh(x_{1}),\ldots,Jg(x_{k})-Jh(x_{k}))} ( M_{0} \varepsilon t ) &=\mu^{k}_{2g(\frac{x_{1}}{2})-2h(\frac{x_{1}}{2}),\ldots,2g( \frac{x_{k}}{2})-2h(\frac{x_{k}}{2}))} ( M_{0}\varepsilon t ) \\ &=\mu^{k}_{g(\frac{x_{1}}{2})-h(\frac{x_{1}}{2}),\ldots,g( \frac{x_{k}}{2})-h(\frac{x_{k}}{2}))} \biggl( \frac{M_{0}}{2} \varepsilon t \biggr) \\ &\geq\frac{\frac{M_{0}}{2}t}{\frac{M_{0}}{2}+\varphi ( \frac{x _{1}}{2},\frac{x_{1}}{2},\ldots,\frac{x_{k}}{2},\frac{x_{k}}{2} ) } \\ &\geq\frac{\frac{M_{0}}{2}t}{\frac{M_{0}}{2}+\frac{M_{0}}{2}\varphi (x_{1},x_{1},\ldots,x_{k},x_{k})} \\ &=\frac{t}{t+\varphi(x_{1},x_{1},\ldots,x_{k},x_{k})} \end{aligned}$$
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
$$ d(J{g},J{h})\leq M_{0}\varepsilon $$
for all \(g,h\in s\). It follows that
$$ \mu_{(f(x_{1})-2f(\frac{x_{1}}{2}),\ldots,f(x_{k})-2f(\frac{x_{k}}{2}))} \biggl( \frac{M_{0}}{2}t \biggr) \geq\frac{t}{t+\varphi (x_{1},x_{1},\ldots,x _{k},x_{k})} $$
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. (1)
    H is a fixed point of J, i.e.,
    $$ H \biggl( \frac{x}{2} \biggr) =\frac{1}{2}H(x) $$
    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. (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. (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
$$\begin{aligned}& \mu^{k}_{(2^{n}(f(\frac{x_{1}}{2^{n}}+\frac{y_{1}}{2^{n}})-f(\frac{x _{1}}{2^{n}})-f(\frac{y_{1}}{2^{n}})),\ldots,2^{n}(f(\frac {x_{k}}{2^{n}}+\frac{y _{k}}{2^{n}})-f(\frac{x_{k}}{2^{n}})-f(\frac{y_{k}}{2^{n}})))}(t) \\& \quad \geq \frac{\frac{t}{2^{n}}}{\frac{t}{2^{n}}+ \frac{M_{0}^{n}}{2^{n}}\varphi(x_{1},y_{1},\ldots,x_{k},y_{k})} \end{aligned}$$
for all \(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k} \in E\), \(t> 0\) and \(n\geq1\). Since
$$ \lim_{n\rightarrow\infty} \frac{\frac{t}{2^{n}}}{\frac{t}{2^{n}}+\frac{M _{0}^{n}}{2^{n}}\varphi(x_{1},y_{1},\ldots,x_{k},y_{k})}=1 $$
for all \(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k} \in E\), \(t> 0\). It follows that
$$ \mu^{k}_{(H(x_{1}+y_{1})-H(x_{1})-H(y_{1}),\ldots,H(x_{k}+y_{k})-H(x_{k})-H(y _{k}))}(t)= 1 $$
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
$$ \mu^{k}_{2^{n}(f(\frac{\beta x_{1}}{2^{n}})-f( \frac{\beta x_{1}}{2^{n}}),\ldots,f(\frac{\beta x_{k}}{2^{n}})-f(\frac{ \beta x_{k}}{2^{n}}))}\bigl(2^{n}t\bigr)\geq \frac{t}{t+\varphi( \frac{x_{1}}{2^{n}},\frac{x_{1}}{2^{n}},\ldots,\frac{x_{k}}{2^{n}},\frac{x _{k}}{2^{n}})} $$
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
$$ \mu^{k}_{2^{n}(f(\frac{\beta x_{1}}{2^{n}})-f( \frac{\beta x_{1}}{2^{n}}),\ldots,f(\frac{\beta x_{k}}{2^{n}})-f(\frac{ \beta x_{k}}{2^{n}}))}(t)\geq\frac{\frac{t}{2^{n}}}{\frac {t}{2^{n}}+\frac{M _{0}^{n}}{2^{n}}\varphi(x_{1},x_{1},\ldots,x_{k},x_{k})} $$
for all \(\beta\in\mathbb{T}\), \(x_{1},\ldots,x_{k} \in E\), \(t> 0\) and \(n\geq1\). We have
$$ \lim_{n\rightarrow\infty}\frac{\frac{t}{2^{n}}}{\frac{t}{2^{n}}+\frac{M _{0}^{n}}{2^{n}}\varphi(x_{1},x_{1},\ldots,x_{k},x_{k})}=1 $$
for all \(x_{1},\ldots,x_{k} \in E\), \(t> 0\), and
$$ \mu^{k}_{(H(\beta x_{1})-\beta H(x_{1}),\ldots,H(\beta x_{k})-\beta H(x _{k}))}(t)=1 $$
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
$$\begin{aligned}& \mu^{k}_{(4^{n}f(\frac{x_{1}}{2^{n}}\frac{y_{1}}{2^{n}})-2^{n}f(\frac{x _{1}}{2^{n}})2^{n}f(\frac{y_{1}}{2^{n}}),\ldots,4^{n} f( \frac{x_{k}}{2^{n}}\frac{y_{k}}{2^{n}})-2^{n}f(\frac {x_{k}}{2^{n}})2^{n}f(\frac{y _{k}}{2^{n}}))}\bigl(4^{n}t\bigr) \\& \quad \geq\frac{t}{t+\varphi(\frac{x_{1}}{2^{n}},\ldots, \frac{x_{k}}{2^{n}},\frac{y_{1}}{2^{n}},\ldots,\frac{y_{k}}{2^{n}})} \end{aligned}$$
for all \(x_{1},\ldots,x_{k} \in E\), \(t> 0\) and \(n\geq1\).
Then we have
$$\begin{aligned}& \mu^{k}_{(4^{n}f(\frac{x_{1}}{2^{n}}\frac{y_{1}}{2^{n}})-2^{n}f(\frac{x _{1}}{2^{n}})2^{n}f(\frac{y_{1}}{2^{n}}),\ldots,4^{n} f( \frac{x_{k}}{2^{n}}\frac{y_{k}}{2^{n}})-2^{n}f(\frac {x_{k}}{2^{n}})2^{n}f(\frac{y _{k}}{2^{n}}))}\bigl(4^{n}t\bigr) \\& \quad \geq\frac{\frac{t}{4^{n}}}{\frac{t}{4^{n}}+\frac{M_{0}^{n}}{t^{n}} \varphi(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k})} \end{aligned}$$
for all \(x_{1},\ldots,x_{k} \in E\), \(t> 0\) and \(n\geq1\).
$$ \lim_{n\rightarrow\infty} \frac{\frac{t}{4^{n}}}{\frac{t}{4^{n}}+\frac{M _{0}^{n}}{t^{n}}\varphi(x_{1},\ldots,x_{k},y_{1},\ldots,y_{k})}=1 $$
for all \(x_{1},\ldots,x_{k} \in E\), \(t> 0\), we have
$$ \mu^{k}_{(H(x_{1}y_{1})-H(x_{1})H(y_{1}),\ldots,H(x_{k}y_{k})-H(x_{k})H(y _{k}))}(t)=1 $$
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:
  1. (i)

    \(\mu_{(\lambda_{1}x_{1},\ldots,\lambda_{k}x_{k})}^{k}(t)= \mu_{(x_{1},\ldots,x_{k})}^{k}(t)\),

  2. (ii)

    \(\mu_{(x_{1},\ldots,x_{k})}^{k}(t)\geq \mu_{(x_{1},\ldots,x_{k},x_{k+1})}^{k+1}(t)\),

  3. (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 \(\alpha+ \beta=1\),

  4. (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
$$ \mu_{(x_{1},\ldots,x_{k})}^{k}(t)\geq\min_{i\in\mathbb{N}_{k}} \mu_{kx _{i}}(t). $$

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
$$ \mu_{ ( f(\frac{x_{1}+y_{1}}{2})-\frac{f(x_{1})}{2}- \frac{f(y_{1})}{2},\ldots,f(\frac{x_{k}+y_{k}}{2})-\frac {f(x_{k})}{2}-\frac{f(y _{k})}{2} ) }^{k} \biggl( \frac{t}{s} \biggr) \geq1-\frac{\alpha}{t}, $$
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
$$ \mu_{(f(x_{1})-T(x_{1}),\ldots,f(x_{k})-T(x_{k}))}^{k} \biggl( \frac {2t}{s} \biggr) \geq1-\frac{\alpha}{t} $$
for all \(x_{1},\ldots,x_{k}\in E\) and \(t,s\in\mathbb{N}\) with \((t,s)=1\).


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
$$ \mu_{(2f(x_{1})-f(2x_{1}),\ldots,2f(x_{k})-f(2x_{k}))}^{k} \biggl( \frac {2t}{s} \biggr) \geq1-\frac{\alpha}{t}. $$
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
$$ \mu_{ ( \frac{f(2^{n-1}x_{1})}{2^{n-1}}- \frac{f(2^{n}x_{1})}{2^{n}},\ldots,\frac{f(2^{n-1}x_{k})}{2^{n-1}}-\frac {f(2^{n}x _{k})}{2^{n}} ) }^{k} \biggl( \frac{t}{2^{n-1}s} \biggr) \geq1-\frac{ \alpha}{2^{n-1}t} $$
for \(n,m\in\mathbb{N}\) with \(n> m\). Using (3.4) and (RN2) we get
$$ \mu_{ ( \frac{f(2^{m}x_{1})}{2^{m}}-\frac {f(2^{n}x_{1})}{2^{n}},\ldots,\frac{f(2^{m}x _{k})}{2^{m}}-\frac{f(2^{n}x_{k})}{2^{n}} ) }^{k} \Biggl( \sum_{i=m} ^{n-1}2^{-i}\frac{t}{s} \Biggr) \geq1-\frac{\alpha}{2^{m}t}. $$
$$ \mu_{ ( \frac{f(2^{m}x_{1})}{2^{m}}-\frac {f(2^{n}x_{1})}{2^{n}},\ldots,\frac{f(2^{m}x _{k})}{2^{m}}-\frac{f(2^{n}x_{k})}{2^{n}} ) }^{k} \biggl( \frac {t}{s} \biggr) \geq1-\frac{\alpha}{2^{m}t} $$
for \(x\in E\). Then, replacing \(x_{1},\ldots,x_{k}\) by \(x,2x,\ldots,2^{k-1}x\) in (3.5), we have
$$ \begin{aligned}[b] & \mu_{ ( \frac{f(2^{m}x)}{2^{m}}-\frac{f(2^{n}x)}{2^{n}},\ldots, \frac{f(2^{m+k-1}x)}{2^{m+k-1}}-\frac{f(2^{n+k-1}x)}{2^{n+k-1}} ) } ^{k} \biggl( \frac{t}{s} \biggr) \\ & \quad \geq 1-\frac{\alpha}{2^{m}t} \\ & \quad \geq 1-\frac{\alpha}{2^{m}}. \end{aligned} $$
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
$$\begin{aligned} \mu_{ ( \frac{f(2^{n}x)}{2^{n}}-\frac{f(2^{n+p}x)}{2^{n+p}},\ldots, \frac{f(2^{n+k-1}x)}{2^{n+k-1}}-\frac {f(2^{n+p+k-1}x)}{2^{n+p+k-1}} ) } ^{k} \biggl( \frac{t}{s} \biggr) & \geq 1-\frac{\alpha}{2^{n}t} \\ &> 1-\varepsilon. \end{aligned}$$
By Lemma 3.1, we have
$$ \mu_{\frac{f(2^{n}x)}{2^{n}}-\frac{f(2^{n+p}x)}{2^{n+p}}} \biggl( \frac {t}{s} \biggr) > 1- \varepsilon $$
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
$$ \mu_{\frac{f(2^{n}x)}{2^{n}}-\frac{f(2^{n+p}x)}{2^{n+p}}}(r)> 1- \varepsilon $$
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
$$ \mu_{ ( \frac{f(2^{n}x_{1})}{2^{n}}-T(x_{1}),\ldots,\frac{f(2^{n}x _{k})}{2^{n}}-T(x_{k}) ) }^{k}(r)\geq\min_{i\in\mathbb{N}_{k}} \mu_{\frac{f(2^{n}x_{i})}{2^{n}}-T(x_{i})} \biggl( \frac{r}{k} \biggr) , $$
$$ \lim_{n\rightarrow\infty} \frac{f(2^{n}x_{k})}{2^{n}} = T(x_{k}). $$
We put \(m= 0\) in (3.5), and we get
$$ \mu_{ ( f(x_{1})-\frac{f(2^{n}x_{1})}{2^{n}},\ldots ,f(x_{k})-\frac{f(2^{n}x _{k})}{2^{n}} ) }^{k} \biggl( \frac{t}{s} \biggr) \geq1-\frac{\alpha }{t}. $$
$$\begin{aligned} &\mu_{f(x_{1})-T(x_{1}),\ldots,f(x_{k})-T(x_{k})}^{k} \biggl( \frac {2t}{s} \biggr) \\ & \quad \geq T_{M} \biggl( \mu_{ ( f(x_{1})-\frac{f(2^{n}x_{1})}{2^{n}},\ldots ,f(x_{k})-\frac{f(2^{n}x _{k})}{2^{n}} ) }^{k} \biggl( \frac{t}{s} \biggr) , \\ & \quad\quad \mu_{ ( \frac{f(2^{n}x_{1})}{2^{n}}-T(x_{1}),\ldots,\frac{f(2^{n}x _{k})}{2^{n}}-T(x_{k}) ) }^{k} \biggl( \frac{t}{s} \biggr) \biggr) \\ & \quad \geq 1-\frac{\alpha}{t} \end{aligned}$$
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
$$ \mu_{ ( f(2^{n}\frac{x+y}{2})-\frac{f(2^{n}x)}{2}- \frac{f(2^{n}y)}{2},\ldots,f(2^{n}\frac{x+y}{2})-\frac{f(2^{n}x)}{2}- \frac{f(2^{n}y)}{2} ) }^{k} \biggl( \frac{2^{n}t}{s} \biggr) \geq 1- \frac{ \alpha}{2^{n}t}. $$
Using (NF4), we conclude that
$$ \mu_{\frac{f2^{n}(\frac{x+y}{2})}{2^{n}}-\frac{1}{2} \frac{f(2^{n}x)}{2^{n}}-\frac{1}{2}\frac{f(2^{n}y)}{2^{n}}} \biggl( \frac {t}{s} \biggr) \geq1- \frac{\alpha}{2^{n}t}. $$
On the other hand, we obtain that
$$\begin{aligned} \mu_{T(\frac{x+y}{2})-\frac{1}{2}T(x)-\frac{1}{2}T(y)} \biggl( \frac {4t}{s} \biggr) \geq& T_{M} \biggl( \mu_{T(\frac{x+y}{2})- \frac{f(2^{n}(\frac{x+y}{2}))}{2^{n}}} \biggl( \frac{t}{s} \biggr) , \\ & \mu_{\frac{T(x)}{2}-\frac{1}{2}\frac{f(2^{n}x)}{2^{n}}}\biggl(\frac{t}{s}\biggr), \\ & \mu_{\frac{T(y)}{2}-\frac{1}{2}\frac{f(2^{n}y)}{2^{n}}} \biggl( \frac {t}{s} \biggr) , \\ & \mu_{\frac{f(2^{n}(\frac{x+y}{2}))}{2^{n}}-\frac{1}{2} \frac{f(2^{n})}{2^{n}x}-\frac{1}{2}\frac{f(2^{n}y)}{2^{n}}} \biggl( \frac {t}{s} \biggr) \biggr) \\ \geq& 1-\frac{\alpha}{2^{n}} \end{aligned}$$
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.
$$ \mu_{T(\frac{x+y}{2})-\frac{1}{2}T(x)-\frac{1}{2}T(y)}(r)\geq1-\frac{ \alpha}{2^{n}} $$
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 \(x\in E\),
$$\begin{aligned} \mu_{T'(x)-T(x)} \biggl( \frac{t}{s} \biggr) &= \mu_{\frac{T'(2^{n}x)}{2^{n}}-\frac{T(2^{n}x)}{2^{n}}} \biggl( \frac {t}{s} \biggr) \\ &\geq T_{M} \biggl( \mu_{T'(2^{n}x)-f(2^{n}x)} \biggl( \frac {2^{n-1}t}{s} \biggr) , \mu_{T(2^{n}x)-f(2^{n}x)} \biggl( \frac{2^{n-1}t}{s} \biggr) \biggr) \\ &\geq1-\frac{\alpha}{2^{n-2}t} \\ &\geq1-\frac{\alpha}{2^{n-2}}. \end{aligned}$$

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.



The authors are grateful to the reviewer(s) for their valuable comments and suggestions.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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

Department of Mathematics, Texas A & M University - Kingsville
Department of Mathematics, Iran University of Science and Technology


  1. Dales, HG, Polyakov, ME: Multi-Normed Spaces. Dissertationes Math. (Rozprawy Mat.), vol. 488 (2012) MATHGoogle Scholar
  2. Saadati, R: Random Operator Theory. Elsevier, Amsterdam (2016) MATHGoogle Scholar
  3. Cho, YJ, Rassias, TM, Saadati, R: Stability of Functional Equations in Random Normed Spaces. Springer Optimization and Its Applications, vol. 86. Springer, New York (2013) MATHGoogle Scholar
  4. Lee, SJ, Saadati, R: On stability of functional inequalities at random lattice ϕ-normed spaces. J. Comput. Anal. Appl. 15(8), 1403-1412 (2013) MathSciNetMATHGoogle Scholar
  5. Vahidi, J, Park, C, Saadati, R: A functional equation related to inner product spaces in non-Archimedean \(\mathcal{L}\)-random normed spaces. J. Inequal. Appl. 2012, 168 (2012) MathSciNetView ArticleMATHGoogle Scholar
  6. Kang, JI, Saadati, R: Approximation of homomorphisms and derivations on non-Archimedean random Lie \(C^{\ast}\)-algebras via fixed point method. J. Inequal. Appl. 2012, 251 (2012) MathSciNetView ArticleMATHGoogle Scholar
  7. Park, C, Eshaghi Gordji, M, Saadati, R: Random homomorphisms and random derivations in random normed algebras via fixed point method. J. Inequal. Appl. 2012, 194 (2012) MathSciNetView ArticleMATHGoogle Scholar
  8. Rassias, JM, Saadati, R, Sadeghi, Gh, Vahidi, J: On nonlinear stability in various random normed spaces. J. Inequal. Appl. 2011, 62 (2011) MathSciNetView ArticleMATHGoogle Scholar
  9. Cho, YJ, Saadati, R: Lattictic non-Archimedean random stability of ACQ functional equation. Adv. Differ. Equ. 2011, 31 (2011) MathSciNetView ArticleMATHGoogle Scholar
  10. Mihet, D, Saadati, R: On the stability of the additive Cauchy functional equation in random normed spaces. Appl. Math. Lett. 24(12), 2005-2009 (2011) MathSciNetView ArticleMATHGoogle Scholar
  11. Mihet, D, Saadati, R, Vaezpour, SM: The stability of the quartic functional equation in random normed spaces. Acta Appl. Math. 110(2), 797-803 (2010) MathSciNetView ArticleMATHGoogle Scholar
  12. Mohamadi, M, Cho, YJ, Park, C, Vetro, P, Saadati, R: Random stability on an additive-quadratic-quartic functional equation. J. Inequal. Appl. 2010, Article ID 754210 (2010) MathSciNetMATHGoogle Scholar
  13. Saadati, R, Vaezpour, SM, Cho, YJ: A note to paper ‘On the stability of cubic mappings and quartic mappings in random normed spaces’. J. Inequal. Appl. 2009, Article ID 214530 (2009) MATHGoogle Scholar
  14. Wang, Z, Sahoo, PK: Stability of an ACQ-functional equation in various matrix normed spaces. J. Nonlinear Sci. Appl. 8(1), 64-85 (2015) MathSciNetMATHGoogle Scholar
  15. Yao, Z: Uniqueness and global exponential stability of almost periodic solution for hematopoiesis model on time scales. J. Nonlinear Sci. Appl. 8(2), 142-152 (2015) MathSciNetMATHGoogle Scholar
  16. Zaharia, C: On the probabilistic stability of the monomial functional equation. J. Nonlinear Sci. Appl. 6(1), 51-59 (2013) MathSciNetMATHGoogle Scholar
  17. Shen, Y, Lan, Y: On the general solution of a quadratic functional equation and its Ulam stability in various abstract spaces. J. Nonlinear Sci. Appl. 7(6), 368-378 (2014) MathSciNetMATHGoogle Scholar
  18. Wang, F, Shen, Y: On the Ulam stability of a quadratic set-valued functional equation. J. Nonlinear Sci. Appl. 7(5), 359-367 (2014) MathSciNetMATHGoogle Scholar
  19. Lu, G, Xie, J, Liu, Q, Jin, Y: Hyers-Ulam stability of derivations in fuzzy Banach space. J. Nonlinear Sci. Appl. 9(12), 5970-5979 (2016) MathSciNetGoogle Scholar
  20. Shen, Y, Chen, W: On the Ulam stability of an n-dimensional quadratic functional equation. J. Nonlinear Sci. Appl. 9(1), 332-341 (2016) MathSciNetMATHGoogle Scholar
  21. Bae, J-H, Park, W-G: On the Ulam stability of the Cauchy-Jensen equation and the additive-quadratic equation. J. Nonlinear Sci. Appl. 8(5), 710-718 (2015) MathSciNetMATHGoogle Scholar
  22. Shen, Y: An integrating factor approach to the Hyers-Ulam stability of a class of exact differential equations of second order. J. Nonlinear Sci. Appl. 9(5), 2520-2526 (2016) MathSciNetMATHGoogle Scholar
  23. Li, T, Zada, A, Faisal, S: Hyers-Ulam stability of nth order linear differential equations. J. Nonlinear Sci. Appl. 9(5), 2070-2075 (2016) MathSciNetMATHGoogle Scholar
  24. Diaz, J, Margolis, B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305-309 (1968) MathSciNetView ArticleMATHGoogle Scholar
  25. Cădariu, L, Radu, V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43-52 (2004) MathSciNetMATHGoogle Scholar
  26. Cădariu, L, Radu, V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Article ID 749392 (2008) MathSciNetMATHGoogle Scholar
  27. Parnami, JC, Vasudeva, HL: On Jensen’s functional equation. Aequ. Math. 43(2-3), 211-218 (1992) MathSciNetView ArticleMATHGoogle Scholar


© The Author(s) 2017