Abstract
By a stochastic controller, we make stable the pseudo stochastic Lie bracket (derivation, derivation) in complex MB-algebras. Next, we get an approximation by a stochastic Lie bracket (derivation, derivation) and calculate the maximum error of the estimate.
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1 Introduction
Let \((\varOmega , \mathfrak{T}, \mu )\) be a probability measure space. Assume that \((T,{\mathfrak{B}}_{T})\) is a Borel measureable space, in which T is an MB-space and \(G,H:\varOmega \times T \to T\) are random derivations. In MB-spaces, first we solve the (additive, additive)–\((\omega ,\nu )\) random operator inequality
where ω, ν are fixed nonzero complex numbers. By a stochastic controller we make stable the pseudo stochastic Lie bracket (derivation, derivation) in complex MB-algebras, associated to the above (additive, additive)–\((\omega ,\nu )\) random operator inequality and the following random operator inequality:
The mentioned process is said to show Hyers–Ulam stability for the (additive, additive)–\((\omega ,\nu )\) random operator inequality (1.1).
2 Preliminaries
Let \(\varXi ^{+}\) be the set of distribution mappings, i.e., the set of all mappings \(\rho :{\mathbb{R}} \cup \{-\infty ,\infty \} \to [0,1]\), writing \(\rho _{\tau }\) for \(\rho (\tau )\), such that ρ is left continuous and increasing on \(\mathbb{R}\). \(O^{+}\subseteq \varXi ^{+}\) includes all mappings \(\rho \in \varXi ^{+}\) for which \(\ell ^{-}\rho _{+\infty }\) is one and \(\ell ^{-}\rho _{\tau }\) is the left limit of the mapping ρ at the point τ, i.e., \(\ell ^{-}\rho _{\tau }=\lim_{\sigma \to \tau ^{-}}\rho _{\sigma }\).
In \(\varXi ^{+}\), we define “≤” as follows:
for each τ in \(\mathbb{R}\) (partially ordered). Note that the function \(\vartheta ^{u}\) defined by
is an element of \(\varXi ^{+}\) and \(\vartheta ^{0}\) is the maximal element in this space (for details, see [1–3]).
Definition 2.1
Denote by I the interval \([0, 1]\). A continuous triangular norm (shortly, a ct-norm) is a continuous binary operation ∗ from \(I^{2}\) to I such that
- (a)
\(\varsigma \ast \tau = \tau \ast \varsigma \) and \(\varsigma \ast (\tau \ast \upsilon ) = ( \varsigma \ast \tau )\ast \upsilon \) for all \(\varsigma ,\tau ,\upsilon \in [0,1]\);
- (b)
\(\varsigma \ast 1=\varsigma \) for all \(\varsigma \in I\);
- (c)
\(\varsigma \ast \tau \leq \upsilon \ast \iota \) whenever \(\varsigma \leq \upsilon \) and \(\tau \leq \iota \) for all \(\varsigma ,\tau ,\upsilon ,\iota \in I\).
Some examples of ct-norms are as follows:
- (1)
\(\varsigma \ast _{P}\tau =\varsigma \tau \);
- (2)
\(\varsigma \ast _{M}\tau =\min \{\varsigma ,\tau \}\);
- (3)
\(\varsigma \ast _{L}\tau =\max \{\varsigma +\tau -1,0\}\) (the Lukasiewicz t-norm).
Definition 2.2
([2])
Suppose that ∗ is a ct-norm, V is a linear space and ξ is a function from V to \(O^{+}\). The ordered tuple \((V,\xi ,\ast )\) is called a Menger normed space (in short, MN-space) if the following conditions are satisfied:
- (MN1)
\(\xi ^{v}_{t}=\vartheta ^{0}_{t}\) for all \(t>0\) if and only if \(v=0\);
- (MN2)
\(\xi ^{\alpha v}_{t}=\xi ^{v}_{\frac{t}{|\alpha |}}\) for all \(v\in V\) and \(\alpha \in \mathbb{C}\) with \(\alpha \neq 0\);
- (MN3)
\(\xi ^{u+v}_{t+s}\geq \xi ^{u}_{t}\ast \xi ^{v}_{s} \) for all \(u,v\in V\) and \(t,s \geq 0\).
A complete MN-space is called Menger Banach space, in short, MB-space. Let \((V,\|\cdot \|)\) be a normed space. Then
defines a Menger norm and the ordered tuple \((V,\xi ,\ast _{M})\) is an MN-space. Also,
defines a Menger norm and the ordered tuple \((V,\xi ,\ast _{M})\) is an MN-space.
Definition 2.3
A Menger normed algebra (in short, MN-algebra) \((V,\xi ,\ast ,\star )\) is an MN-space \((V,\xi ,\ast )\) with algebraic structure such that
- (FN-5)
\(\xi ^{uv}_{ts}\geq \xi ^{u}_{t}\star \xi ^{v}_{s}\) for all \(u,v\in V\) and all \(t,s> 0\). in which ⋆ is a ct-norm.
Every normed algebra \((V,\|\cdot \|)\) defines an MN-algebra \((V,\xi ,\ast _{M},\ast _{P})\), where
if and only if
This space is called the induced MN-algebra. A complete MN-algebra is called Menger Banach algebra, in short, MB-algebra. Let \((\varGamma , \varSigma , \xi )\) be a probability measure space. Assume that \((T,{\mathfrak{B}}_{T})\) and \((S,{\mathfrak{B}}_{S})\) are Borel measurable spaces, in which T and S are complete MN-spaces. A mapping \(F:\varGamma \times T\to S\) is said to be a random operator if \(\{\gamma : F(\gamma ,t)\in B\}\in \varSigma \) for all t in T and \(B\in {\mathfrak{B}}_{S}\). Also, F is a random operator if \(F(\gamma ,t)=s(\gamma )\) is an S-valued random variable for all t in T. A random operator \(F:\varGamma \times T\to S\) is called linear if \(F(\gamma ,\alpha t_{1}+\beta t_{2})=\alpha F(\gamma ,t_{1})+ \beta F( \gamma , t_{2})\) almost everywhere for \(t_{1}, t_{2} \in T\) and α, β scalars, and bounded if there is a nonnegative random variable \(M(\gamma )\) such that
almost everywhere for each \(t,s\in T\) and \(\tau >0\).
Let T be an MB-algebra. A linear random operator \(\pi :\varGamma \times T\to T\) that satisfies
for all \(t,s\in T\) and \(\gamma \in \varGamma \), is called stochastic derivation.
We denote by \(\varPi (\varGamma ,T)\) the set of \(\mathbb{C}\)-linear bounded stochastic derivations on \(\varGamma \times T\). For \(\pi _{1},\pi _{2}\in \varPi (\varGamma ,T)\),
for all \(t,s\in T\) and \(\gamma \in \varGamma \). Assume that \([\pi _{1},\pi _{2}]=\pi _{1}o\pi _{2}-\pi _{2}o\pi _{1}\). Then
for all \(t,s\in T\) and \(\gamma \in \varGamma \). The \(\mathbb{C}\)-linearity of \([\pi _{1},\pi _{2}]\) implies that \([\pi _{1},\pi _{2}]\in \varPi ( \varGamma ,T)\) for all \(\pi _{1},\pi _{2}\in \varPi (\varGamma ,T)\). Then \(\varPi (\varGamma ,T)\) is a stochastic Lie algebra with stochastic Lie bracket \([\pi _{1},\pi _{2}]\), \(\pi _{1}+\pi _{2}\) and \(\beta \pi _{1}\) are \(\mathbb{C}\)-linear stochastic derivations in which \(\beta \in \mathbb{C}\).
Definition 2.4
Consider an MB-algebra T and linear random operators \(\varTheta ,\varPhi :\varGamma \times T\to T\). Set \([\varTheta ,\varPhi ](\gamma ,t)=\varTheta (\gamma ,\varPhi (\gamma ,t))-\varPhi ( \gamma ,\varTheta (\gamma ,t))\) for every \(t\in T\) and \(\gamma \in \varGamma \). The linear operator \([\varTheta ,\varPhi ]:\varGamma \times T\to T\) is said a stochastic Lie bracket (derivation, derivation) when
for all \(t,s\in T\) and \(\gamma \in \varGamma \).
Recently, some authors have published some papers on approximation of functional equations in various spaces by the direct technique and the fixed point technique, for example, fuzzy Menger normed algebras [5], fuzzy metric spaces [7], fuzzy normed spaces [8], non-Archimedian random Lie \(C^{*}\)-algebras [9], random multi-normed space [10], non-Archimedean random normed spaces [6]; see also [11–30].
Note that a \([0,\infty ]\)-valued metric is called a generalized metric.
Theorem 2.5
Consider a complete generalized metric space\((T, \delta )\)and a strictly contractive function\(\varLambda : T \rightarrow T\)with Lipschitz constant\(\beta <1\). Then, for every given element\(t\in T\), either
for each\(n\in \mathbb{N}\)or there is an\(n_{0}\in \mathbb{N}\)such that
- (1)
\(\delta (\varLambda ^{n}t,\varLambda ^{n+1}t)<\infty \), for all\(n \ge n_{0}\);
- (2)
the sequence\(\{ \varLambda ^{n} t\}\)converges to a fixed point\(s^{*}\)ofΛ;
- (3)
\(s^{*}\)is the unique fixed point ofΛin the set\(V = \{s\in T \mid \delta (\varLambda ^{n_{0}}t,s) <\infty \}\);
- (4)
\((1-\beta )\delta (s,s^{\ast }) \le \delta (s,\varLambda s)\)for every\(s \in V\).
3 Stability of (additive, additive) \((\omega ,\nu )\)-random operator inequality: direct technique
Hereinafter we suppose that \(\ast =\ast _{M}\).
Lemma 3.1
Assume that random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\)and
for all\(t,s\in T\), \(\gamma \in \varGamma \)and\(\tau >0\)in which\(\vert \nu \vert <1\)and\(\vert \omega \vert <1\). Then the random operators\(G,H:\varGamma \times T \to T\)are additive.
Proof
Putting \(s=t\) in (3.1), we get
for all \(t\in T\) and \(\gamma \in \varGamma \). Then \(G(\gamma ,2t)=2G(\gamma ,t)\) and \(H(\gamma ,2t)=2H(\gamma ,t)\) for all \(t\in T\) and \(\gamma \in \varGamma \). By (3.1) we have
for all \(t,s\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). So \(\vert \nu \vert <1\) and \(\vert \omega \vert <1\) imply that \(G(\gamma ,t+s)-G(\gamma ,t)-G(\gamma ,s)=0\) and \(H(\gamma ,t+s)+H(\gamma ,t-s)-2H(\gamma ,t)=0\) for all \(t\in T\) and \(\gamma \in \varGamma \). Thus the random operators \(G,H:\varGamma \times T \to T\) are additive. □
Lemma 3.2
([34, Theorem 2.1])
Assume that a random operator\(F:\varGamma \times T \to T\)is additive and
for all\(d\in \mathbb{D}^{1}:=\{c\in \mathbb{C}:\vert c\vert =1\}\)and each\(t\in T\)and\(\gamma \in \varGamma \). Then the random operator\(F:\varGamma \times T \to T\)is\(\mathbb{C}\)-linear.
Theorem 3.3
Let\((T,\xi ,\ast ,\ast )\)be an MB-algebra. Let\(\varphi : T^{2}\to O^{+}\)be a distribution function such that there exists a\(\beta \in (0,1)\)with
for all\(t,s\in T\)and\(\tau >0\). Suppose that random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\)and
for all\(d\in \mathbb{D}^{1}\), \(t,s\in T\), \(\gamma \in \varGamma \)and\(\tau >0\). Assume that the random operators\(G,H:\varGamma \times T \to T\)satisfy
for all\(t,s\in T\), \(\gamma \in \varGamma \)and\(\tau >0\). Then there are a unique\(\mathbb{C}\)-linear random operator\(\varTheta :\varGamma \times T \to T\)and a unique stochastic derivation\(\pi :\varGamma \times T \to T\)such that\([\varTheta ,\pi ]:\varGamma \times T \to T\)is a stochastic derivation and
for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).
Proof
In (3.3), putting \(d=1\) and \(s=t\), one obtains
and so
for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). Replacing t by \(\frac{t}{2^{n}}\) in (3.7), we get
for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(n\in \mathbb{N}\). Since
we have
and so
for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(n\in \mathbb{N}\).
Replacing t by \(\frac{t}{2^{m}}\) in (3.10), we get
for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(n,m\in \mathbb{N}\).
Let \(m,n\to \infty \) in (3.11), since \(\beta \in (0,1)\), we conclude that \(\varphi ^{t,t}_{ \frac{\tau }{\sum _{k=m+1}^{n+m}\frac{1}{2}\beta ^{k}}}\) tends to 1 for all \(\tau >0\). Thus this shows that \(\{2^{n}G(\gamma ,\frac{t}{2^{n}})\}\) and \(\{2^{n}H(\gamma ,\frac{t}{2^{n}})\}\) are Cauchy sequences for each \(t\in T\), \(\gamma \in \varGamma \). Since T is complete, the mentioned sequences converge. Now we define the random operators \(\varTheta ,\pi :\varGamma \times T \to T\) by
for each \(t\in T\), \(\gamma \in \varGamma \). Putting \(m=0\) and \(n\to +\infty \) in (3.11), we obtain (3.5).
Using (3.3), (3.12) and letting n tend to +∞, we have
for all \(d\in \mathbb{D}^{1}\), \(t,s\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). Then
for all \(d\in \mathbb{D}^{1}\) and \(t,s\in T\), \(\gamma \in \varGamma \), \(\tau >0\). Putting \(d=1\) in (3.13) and using Lemma 3.1, we see that the random operators \(\varTheta ,\pi :\varGamma \times T \to T\) are additive.
The additivity of Θ and π and (3.13) imply that
for all \(d\in \mathbb{D}^{1}\) and \(t,s\in T\), \(\gamma \in \varGamma \), \(\tau >0\), which implies that
Then \(\varTheta (\gamma ,d t)=d \varTheta (\gamma ,t)\) and \(\pi (\gamma ,d t)=d \pi (\gamma ,t)\) for all \(d\in \mathbb{D}^{1}\) and \(t\in T\), \(\gamma \in \varGamma \). Now, Lemma 3.2 implies that the additive mappings Θ and π are \(\mathbb{C}\)-linear.
The additivity of Θ and π and (3.4) imply that
which tends to 1 as \(n\to +\infty \). Then
for all \(t,s\in T\), \(\gamma \in \varGamma \). Thus \([\varTheta ,\phi ]\) and π are stochastic derivations. □
Corollary 3.4
Let\((T,\xi ,\ast ,\ast )\)be an MB-algebra. Assume that\(q>0\)and\(p>1\). Suppose that random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\)and
for all\(d\in \mathbb{D}^{1}\), \(t,s\in T\), \(\gamma \in \varGamma \)and\(\tau >0\). Let
for all\(t,s\in T\), \(\gamma \in \varGamma \)and\(\tau >0\). Then there are a unique\(\mathbb{C}\)-linear random operator\(\varTheta :\varGamma \times T \to T\)and a unique stochastic derivation\(\pi :\varGamma \times T \to T\)such that\([\varTheta ,\pi ]:\varGamma \times T \to T\)is a stochastic derivation and
for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).
Proof
In Theorem 3.3, putting
and letting \(\beta =2^{1-p}\), we get the desired result. □
Theorem 3.5
Let\((T,\xi ,\ast ,\ast )\)be an MB-algebra. Let\(\varphi : T^{2}\to O^{+}\)be a distribution function such that there exists a\(\beta \in (0,1)\)with
for all\(t,s\in T\)and\(\tau >0\). Suppose that the random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\), (3.3) and (3.4). Then there are a unique\(\mathbb{C}\)-linear random operator\(\varTheta :\varGamma \times T \to T\)and a unique stochastic derivation\(\pi :\varGamma \times T \to T\)such that\([\varTheta ,\pi ]:\varGamma \times T \to T\)is a stochastic derivation and
for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).
Proof
Using (3.6), we get
for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\).
Replacing t by \(2^{n}t\) in (3.21), we get
for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(n\in \mathbb{N}\). Since
we have
and so
for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(n\in \mathbb{N}\).
Replacing t by \(2^{m}t\) in (3.24), we get
for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(n,m\in \mathbb{N}\).
Letting \(m,n\rightarrow + \infty \) in (3.25), since \(\beta \in (0,1)\), we conclude that \(\varphi ^{t,t}_{ \frac{\tau }{\sum _{k=m}^{n+m}\frac{(4\beta )^{k}}{2^{k+1}}}}\) tends to 1 for all \(\tau >0\). This shows that \(\{\frac{1}{2^{n}}G(\gamma ,2^{n}t)\}\) and \(\{\frac{1}{2^{n}}H(\gamma ,2^{n}t)\}\) are Cauchy sequences for each \(t\in T\), \(\gamma \in \varGamma \). Since T is complete, the mentioned sequences converge. Now we define the random operators \(\varTheta ,\pi :\varGamma \times T \to T\) by
for each \(t\in T\), \(\gamma \in \varGamma \). Putting \(m=0\) and \(n\to \infty \) in (3.25), we get (3.5). By the same method in the proof of Theorem 3.3, the random operators \(\varTheta ,\pi :\varGamma \times T\rightarrow T\) are \(\mathbb{C}\)-linear.
The additivity of Θ and π and (3.4) imply that
which tends to 1 as \(n\rightarrow +\infty \). Then
for all \(t,s\in T\), \(\gamma \in \varGamma \). Thus \([\varTheta ,\phi ]\) and π are stochastic derivations. □
Corollary 3.6
Let\((T,\xi ,\ast ,\ast )\)be an MB-algebra. Assume that\(q>0\)and\(p<1\). Suppose that random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\), (3.16) and (3.17). Then there are a unique\(\mathbb{C}\)-linear random operator\(\varTheta :\varGamma \times T \to T\)and a unique stochastic derivation\(\pi :\varGamma \times T \to T\)such that\([\varTheta ,\pi ]:\varGamma \times T \to T\)is a stochastic derivation and
for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).
Proof
In Theorem 3.5, putting
and letting \(\beta =2^{p-1}\), we get the desired result. □
4 Stability of (additive, additive) \((\omega ,\nu )\)-random operator inequality (1.1) via fixed point technique
Theorem 4.1
Let\((T,\xi ,\ast ,\ast )\)be an MB-algebra. Let\(\varphi : T^{2}\to O^{+}\)be a distribution function such that there exists a\(\beta \in (0,1)\)with
for all\(t,s\in T\)and\(\tau >0\). Suppose that random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\), (3.3) and (3.4). Then there are a unique\(\mathbb{C}\)-linear random operator\(\varTheta :\varGamma \times T \to T\)and a unique stochastic derivation\(\pi :\varGamma \times T \to T\)such that\([\varTheta ,\pi ]:\varGamma \times T \to T\)is a stochastic derivation and
for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).
Proof
By Theorem 3.3, there exist a unique \(\mathbb{C}\)-linear random operator \(\varTheta :\varGamma \times T \to T\) and a unique stochastic derivation \(\pi :\varGamma \times T \to T\) such that \([\varTheta ,\pi ]: \varGamma \times T \to T\) is a stochastic a derivation.
In (3.3), putting \(d=1\) and \(s=t\), we get
and so
for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\).
On the set
we define the following generalized metric on S:
In [35], Miheţ and Radu proved that \((S, \delta )\) is complete (see also [36]).
Now, we consider the linear mapping \(\varLambda :S\to S\) such that
for all \(t\in T\), \(\gamma \in \varGamma \).
Let \((G,H),(G_{1},H_{1})\in S\) be given such that \(\delta ((G,H),(G_{1},H_{1}))=\varepsilon \). Then
for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). So
for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(\delta (\varLambda (G,H),\varLambda (G_{1},H_{1}))\leq \beta \varepsilon \). This means that
for all \((G,H),(G_{1},H_{1})\in S\).
It follows from (3.3) that
for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). So \(\delta ((G,H),\varLambda (G,H))\leq \frac{\beta }{2}\). By Theorem 2.5, there exist random operators \(\varTheta ,\pi :\varGamma \times T\rightarrow T\) satisfying the following:
(1) There is a fixed point \((\varTheta ,\pi )\) for the function Λ such that
for all \(t\in T\), \(\gamma \in \varGamma \). The random operator \((\varTheta ,\pi )\) is a unique fixed point of Λ in the set
(2) \(\delta (\varLambda ^{n}(G,H),(\varTheta ,\pi ))\to 0\) as \(n\rightarrow +\infty \). which implies
(3) \(\delta ((G,H),(\varTheta ,\pi ))\leq \frac{1}{1-\beta }\delta ((G,H), \varLambda (G,H))\), which implies
for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). □
Corollary 4.2
Let\((T,\xi ,\ast ,\ast )\)be an MB-algebra. Assume that\(q>0\)and\(p>1\). Suppose that random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\), (3.16) and (3.17). Then there are a unique\(\mathbb{C}\)-linear random operator\(\varTheta :\varGamma \times T \to T\)and a unique stochastic derivation\(\pi :\varGamma \times T \to T\)such that\([\varTheta ,\pi ]:\varGamma \times T \to T\)is a stochastic derivation and
for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).
Proof
In Theorem 4.1, putting
and letting \(\beta =2^{1-p}\), we get the desired result. □
Theorem 4.3
Let\((T,\xi ,\ast ,\ast )\)be an MB-algebra. Let\(\varphi : T^{2}\to O+\)be a distribution function such that there exists a\(\beta \in (0,1)\)with
for all\(t,s\in T\)and\(\tau >0\). Suppose that random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\), (3.3) and (3.4). Then there are a unique\(\mathbb{C}\)-linear random operator\(\varTheta :\varGamma \times T \to T\)and a unique stochastic derivation\(\pi :\varGamma \times T \to T\)such that\([\varTheta ,\pi ]:\varGamma \times T \to T\)is a stochastic derivation and
for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).
Proof
By Theorem 3.5, there exist a unique \(\mathbb{C}\)-linear random operator \(\varTheta :\varGamma \times T \to T\) and a unique stochastic derivation \(\pi :\varGamma \times T \to T\) such that \([\varTheta ,\pi ]: \varGamma \times T \to T\) is a stochastic a derivation.
Let \((S,\delta )\) be the generalized metric space defined in the proof of Theorem 4.1. Now, we consider the linear mapping \(\varLambda :S\to S\) such that
for all \(t\in T\), \(\gamma \in \varGamma \). It follows from (4.3) that
for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). The proof will be finished by a similar method to the one used in the proofs of Theorems 3.3 and 4.1. □
Corollary 4.4
Let\((T,\xi ,\ast ,\ast )\)be an MB-algebra. Assume that\(q>0\)and\(p<1\). Suppose that random operators\(G,H:\varGamma \times T \to T\)satisfy\(G(\gamma ,0)=H(\gamma ,0)=0\), (3.16) and (3.17). Then there are a unique\(\mathbb{C}\)-linear random operator\(\varTheta :\varGamma \times T \to T\)and a unique stochastic derivation\(\pi :\varGamma \times T \to T\)such that\([\varTheta ,\pi ]:\varGamma \times T \to T\)is a stochastic derivation and
for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).
Proof
In Theorem 4.3, putting
and letting \(\beta =2^{p-1}\), we get the desired result. □
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This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937).
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Madadi, M., Saadati, R., Park, C. et al. Stochastic Lie bracket (derivation, derivation) in MB-algebras. J Inequal Appl 2020, 141 (2020). https://doi.org/10.1186/s13660-020-02407-8
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DOI: https://doi.org/10.1186/s13660-020-02407-8