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Stochastic Lie bracket (derivation, derivation) in MB-algebras

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.

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

$$\begin{aligned}& \xi ^{G(\gamma ,t+s)-G(\gamma ,t)-G(\gamma ,s)}_{\tau } *\xi ^{H( \gamma ,t+s)+H(\gamma ,t-s)-2H(\gamma ,t)}_{\tau } \\& \quad \geq \xi ^{\omega (2G(\gamma ,\frac{t+s}{2})-G(\gamma ,t)-G( \gamma ,s))}_{\tau } * \xi ^{\nu (2H(\gamma ,\frac{t+s}{2})+2H( \gamma ,\frac{t-s}{2})-2H(\gamma ,t))}_{\tau }, \end{aligned}$$
(1.1)

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:

$$ \xi ^{[G,H] (\gamma ,ts)-[G,H](\gamma ,t)s-t[G,H](\gamma ,s)}_{\tau } \ast \xi ^{H(\gamma ,ts)-H(\gamma ,t)s-tH(\gamma ,s)}_{\tau }\geq \varphi ^{t,s}_{\tau }. $$
(1.2)

The mentioned process is said to show Hyers–Ulam stability for the (additive, additive)–\((\omega ,\nu )\) random operator inequality (1.1).

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:

$$ \rho \leq \varrho \quad \text{if and only if}\quad \rho _{\tau }\leq \varrho _{\tau } $$

for each τ in \(\mathbb{R}\) (partially ordered). Note that the function \(\vartheta ^{u}\) defined by

$$ \vartheta ^{u}_{s}= \textstyle\begin{cases} 0, & \text{if } s\leq u, \\ 1, & \text{if } s>u, \end{cases} $$

is an element of \(\varXi ^{+}\) and \(\vartheta ^{0}\) is the maximal element in this space (for details, see [13]).

Definition 2.1

([1, 4])

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

  1. (a)

    \(\varsigma \ast \tau = \tau \ast \varsigma \) and \(\varsigma \ast (\tau \ast \upsilon ) = ( \varsigma \ast \tau )\ast \upsilon \) for all \(\varsigma ,\tau ,\upsilon \in [0,1]\);

  2. (b)

    \(\varsigma \ast 1=\varsigma \) for all \(\varsigma \in I\);

  3. (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. (1)

    \(\varsigma \ast _{P}\tau =\varsigma \tau \);

  2. (2)

    \(\varsigma \ast _{M}\tau =\min \{\varsigma ,\tau \}\);

  3. (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:

  1. (MN1)

    \(\xi ^{v}_{t}=\vartheta ^{0}_{t}\) for all \(t>0\) if and only if \(v=0\);

  2. (MN2)

    \(\xi ^{\alpha v}_{t}=\xi ^{v}_{\frac{t}{|\alpha |}}\) for all \(v\in V\) and \(\alpha \in \mathbb{C}\) with \(\alpha \neq 0\);

  3. (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

$$ \xi ^{v}_{s}= \textstyle\begin{cases} 0, & \text{if } s\leq 0, \\ \exp (-\frac{ \Vert v \Vert }{s}), & \text{if } s>0, \end{cases} $$

defines a Menger norm and the ordered tuple \((V,\xi ,\ast _{M})\) is an MN-space. Also,

$$ \xi ^{v}_{s}= \textstyle\begin{cases} 0, & \text{if } s\leq 0, \\ \frac{s}{s+ \Vert v \Vert }, & \text{if } s>0, \end{cases} $$

defines a Menger norm and the ordered tuple \((V,\xi ,\ast _{M})\) is an MN-space.

Definition 2.3

([5, 6])

A Menger normed algebra (in short, MN-algebra) \((V,\xi ,\ast ,\star )\) is an MN-space \((V,\xi ,\ast )\) with algebraic structure such that

  1. (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

$$ \xi ^{v}_{s}= \textstyle\begin{cases} 0, & \text{if } s\leq 0, \\ \exp (-\frac{ \Vert v \Vert }{s}), & \text{if } s>0, \end{cases} $$

if and only if

$$ \Vert uv \Vert \le \Vert u \Vert \Vert v \Vert + s \Vert v \Vert + t \Vert u \Vert \quad (u,v \in V; t,s > 0). $$

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

$$ \xi ^{F(\gamma ,t)-F(\gamma ,s)}_{M(\gamma )\tau }\ge \xi ^{t-s}_{ \tau } $$

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

$$ \pi (\gamma ,ts)=\pi (\gamma ,t)s+t\pi (\gamma ,s) $$

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 \(\pi _{1},\pi _{2}\in \varPi (\varGamma ,T)\),

$$\begin{aligned}& \pi _{1}o\pi _{2}(\gamma ,ts)=\pi _{1}o \pi _{2}(\gamma ,t)s+\pi _{2}( \gamma ,t)\pi _{1}(\gamma ,s)+\pi _{1}(\gamma ,t)\pi _{2}(\gamma ,s)+t \pi _{1}o\pi _{2}(\gamma ,s), \\& \pi _{2}o\pi _{1}(\gamma ,ts)=\pi _{2}o \pi _{1}(\gamma ,t)s+\pi _{1}( \gamma ,t)\pi _{2}(\gamma ,s)+\pi _{2}(\gamma ,t)\pi _{1}(\gamma ,t)+t \pi _{2}o\pi _{1}(\gamma ,s), \end{aligned}$$

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

$$ [\pi _{1},\pi _{2}](\gamma ,ts)=[\pi _{1},\pi _{2}](\gamma ,t)s+t[ \pi _{1},\pi _{2}](\gamma ,s) $$

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

$$\begin{aligned}& [\varTheta ,\varPhi ](\gamma ,ts)=[\varTheta ,\varPhi ](\gamma ,t)s+t[\varTheta , \varPhi ]( \gamma ,s), \\& \varPhi (\gamma ,ts)=\varPhi (\gamma ,t)s+t\varPhi (\gamma ,s), \end{aligned}$$

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 [1130].

Note that a \([0,\infty ]\)-valued metric is called a generalized metric.

Theorem 2.5

([3133])

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

$$ \delta \bigl(\varLambda ^{n}t,\varLambda ^{n+1}t\bigr) = \infty $$

for each\(n\in \mathbb{N}\)or there is an\(n_{0}\in \mathbb{N}\)such that

  1. (1)

    \(\delta (\varLambda ^{n}t,\varLambda ^{n+1}t)<\infty \), for all\(n \ge n_{0}\);

  2. (2)

    the sequence\(\{ \varLambda ^{n} t\}\)converges to a fixed point\(s^{*}\)ofΛ;

  3. (3)

    \(s^{*}\)is the unique fixed point ofΛin the set\(V = \{s\in T \mid \delta (\varLambda ^{n_{0}}t,s) <\infty \}\);

  4. (4)

    \((1-\beta )\delta (s,s^{\ast }) \le \delta (s,\varLambda s)\)for every\(s \in V\).

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

$$\begin{aligned}& \xi ^{G(\gamma ,t+s)-G(\gamma ,t)-G(\gamma ,s)}_{\tau }\ast \xi ^{H( \gamma ,t+s)+H(\gamma ,t-s)-2H(\gamma ,t)}_{\tau } \\& \quad \geq \xi ^{ \omega (2G (\gamma ,\frac{t+s}{2} )-G(\gamma ,t)-G(\gamma ,s) )}_{\tau }\ast \xi ^{ \nu (2H (\gamma ,\frac{t+s}{2} )+2H (\gamma , \frac{t-s}{2} )-2H(\gamma ,t) )}_{\tau } \end{aligned}$$
(3.1)

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

$$ \xi ^{G(\gamma ,2t)-2G(\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,2t)-2H( \gamma ,t)}_{\tau } \geq \vartheta ^{0}_{\tau } $$

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

$$\begin{aligned}& \xi ^{G(\gamma ,t+s)-G(\gamma ,t)-G(\gamma ,s)}_{\tau }\ast \xi ^{H( \gamma ,t+s)+H(\gamma ,t-s)-2H(\gamma ,t)}_{\tau } \\& \quad \geq \xi ^{ \omega (G(\gamma ,t+s)-G(\gamma ,t)-G( \gamma ,s) )}_{\tau }\ast \xi ^{ \nu (H (\gamma ,t+s)+ H( \gamma , t-s)-2H(\gamma ,t))}_{\tau } \end{aligned}$$

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

$$ F(\gamma ,dt)=d F(\gamma ,t) $$

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

$$ \varphi ^{\frac{t}{2},\frac{s}{2}}_{\frac{\beta }{2}\tau }\ge \varphi ^{\frac{t}{2},\frac{s}{2}}_{\frac{\beta }{4}\tau }\ge \varphi ^{t,s}_{\tau } $$
(3.2)

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

$$\begin{aligned}& \xi ^{G(\gamma ,d(t+s))-d G(\gamma ,t)-d G(\gamma ,s)}_{ \tau } \ast \xi ^{H(\gamma ,d(t+s))+H(\gamma ,d(t-s))-2d H(\gamma ,t)}_{ \tau } \\& \quad \geq \xi ^{ \omega (2G (\gamma , d\frac{t+s}{2} )-d G(\gamma ,t)-d G(\gamma ,s) )}_{\tau } \\& \qquad {} \ast \xi ^{ \nu (2H (\gamma ,d\frac{t+s}{2} )+2H (\gamma ,d\frac{t-s}{2} )-2d H(\gamma ,t) )}_{\tau }\ast \varphi ^{t,s}_{\tau } \end{aligned}$$
(3.3)

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

$$ \xi ^{[G,H] (\gamma ,ts)-[G,H](\gamma ,t)s-t[G,H](\gamma ,s)}_{\tau } \ast \xi ^{H(\gamma ,ts)-H(\gamma ,t)s-t H(\gamma ,s)}_{\tau }\geq \varphi ^{t,s}_{\tau } $$
(3.4)

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

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau }\geq \varphi ^{t,t}_{ \frac{2(1-\beta )}{\beta }\tau } $$
(3.5)

for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).

Proof

In (3.3), putting \(d=1\) and \(s=t\), one obtains

$$ \xi ^{G(\gamma ,2t)-2G(\gamma ,t)}_{\tau }* \xi ^{H(\gamma ,2t)-2H( \gamma ,t)}_{\tau }\geq \varphi ^{t,t}_{\tau } $$
(3.6)

and so

$$\begin{aligned} \xi ^{G(\gamma ,t)-2G (\gamma ,\frac{t}{2} )}_{\tau }\ast \xi ^{H(\gamma ,t)-2H (\gamma ,\frac{t}{2} )}_{\tau } \geq & \varphi ^{\frac{t}{2},\frac{t}{2}}_{\tau } \\ \geq & \varphi ^{t,t}_{\frac{2}{\beta }\tau } \end{aligned}$$
(3.7)

for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). Replacing t by \(\frac{t}{2^{n}}\) in (3.7), we get

$$\begin{aligned} \xi ^{2^{n} G (\gamma ,\frac{t}{2^{n}} )-2^{n+1}G ( \gamma ,\frac{t}{2^{n+1}} )}_{\tau }\ast \xi ^{2^{n}H ( \gamma ,\frac{t}{2^{n}} )-2^{n+1}H (\gamma , \frac{t}{2^{n+1}} )}_{\tau } \geq & \varphi ^{\frac{t}{2^{n+1}}, \frac{t}{2^{n+1}}}_{\frac{2}{\beta }\tau } \\ \geq & \varphi ^{t,t}_{\frac{2}{\beta ^{n+1}}\tau } \end{aligned}$$
(3.8)

for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(n\in \mathbb{N}\). Since

$$ 2^{n}G \biggl(\gamma ,\frac{t}{2^{n}} \biggr)-G(\gamma ,t)= \sum_{k=1}^{n}2^{k}G \biggl(\gamma ,\frac{t}{2^{k}} \biggr)-2^{k-1}G \biggl(\gamma , \frac{t}{2^{k-1}} \biggr), $$

we have

$$\begin{aligned}& \xi ^{2^{n} G (\gamma ,\frac{t}{2^{n}} )-G (\gamma ,t )}_{\sum _{k=1}^{n}\frac{1}{2}\beta ^{k}\tau }\ast \xi ^{2^{n}H (\gamma ,\frac{t}{2^{n}} )-H (\gamma ,t )}_{ \sum _{k=1}^{n}\frac{1}{2}\beta ^{k}\tau } \\& \quad \geq \prod_{k=1}^{n} \bigl[ \xi ^{2^{k} G (\gamma , \frac{t}{2^{k}} )-2^{k-1}G (\gamma ,\frac{t}{2^{k-1}} )}_{\frac{1}{2}\beta ^{k}\tau }\ast \xi ^{2^{k}H (\gamma , \frac{t}{2^{k}} )-2^{k-1}H (\gamma ,\frac{t}{2^{k-1}} )}_{\frac{1}{2}\beta ^{k}\tau } \bigr] \\& \quad \geq \varphi ^{t,t}_{\tau } \end{aligned}$$
(3.9)

and so

$$ \xi ^{2^{n} G (\gamma ,\frac{t}{2^{n}} )-G (\gamma ,t )}_{\tau }\ast \xi ^{2^{n}H (\gamma ,\frac{t}{2^{n}} )-H (\gamma ,t )}_{\tau } \geq \varphi ^{t,t}_{ \frac{\tau }{\sum _{k=1}^{n}\frac{1}{2}\beta ^{k}}} $$
(3.10)

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

$$\begin{aligned} \xi ^{2^{n+m} G (\gamma ,\frac{t}{2^{n+m}} )-2^{m}G ( \gamma ,\frac{t}{2^{m}} )}_{\tau }\ast \xi ^{2^{n+m}H ( \gamma ,\frac{t}{2^{n+m}} )-2^{m}H (\gamma , \frac{t}{2^{n+m}} )}_{\tau } \geq& \varphi ^{\frac{t}{2^{m}}, \frac{t}{2^{m}}}_{ \frac{2^{m}\tau }{\sum _{k=1}^{n}\frac{1}{2}\beta ^{k}}} \\ \geq& \varphi ^{t,t}_{ \frac{\tau }{\sum _{k=m+1}^{n+m}\frac{1}{2}\beta ^{k}}}, \end{aligned}$$
(3.11)

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

$$ \varTheta (\gamma ,t):=\lim_{n\to +\infty }2^{n} G \biggl(\gamma , \frac{t}{2^{n}} \biggr), \qquad \pi (\gamma ,t):=\lim _{n \to +\infty }2^{n} H \biggl(\gamma , \frac{t}{2^{n}} \biggr) $$
(3.12)

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

$$\begin{aligned}& \xi ^{\varTheta (\gamma ,d(t+s))-d \varTheta (\gamma ,t)-d \varTheta (\gamma ,s)}_{\tau }\ast \xi ^{\pi (\gamma ,d(t+s))+\pi ( \gamma ,d(t-s))-2d \pi (\gamma ,s)}_{\tau } \\& \quad = \xi ^{G(\gamma ,d(\frac{t+s}{2^{n}}))-d G(\gamma , \frac{t}{2^{n}})-d G(\gamma ,\frac{t}{2^{n}})}_{\frac{\tau }{2^{n}}} \ast \xi ^{H(\gamma ,d(\frac{t+s}{2^{n}}))+H(\gamma ,d( \frac{t-s}{2^{n}}))-2d H(\gamma ,\frac{s}{2^{n}})}_{ \frac{\tau }{2^{n}}} \\& \quad \geq \xi ^{ \omega (2G (\gamma , d\frac{t+s}{2^{n+1}} )-d G(\gamma ,\frac{t}{2^{n}})-d G(\gamma ,\frac{s}{2^{n}}) )}_{\frac{\tau }{2^{n}}} \ast \xi ^{ \nu (2H (\gamma ,d\frac{t+s}{2^{n+1}} )+2H (\gamma ,d\frac{t-s}{2^{n+1}} )-2d H(\gamma , \frac{t}{2^{n}}) )}_{\frac{\tau }{2^{n}}}\ast \varphi ^{ \frac{t}{2^{n}},\frac{s}{2^{n}}}_{\frac{\tau }{2^{n}}} \\& \quad \geq \xi ^{ \omega (2\varTheta (\gamma , d\frac{t+s}{2} )-d \varTheta (\gamma ,t)-d \varTheta (\gamma ,s) )}_{\tau } \ast \xi ^{ \nu (2\pi (\gamma ,d\frac{t+s}{2} )+2 \pi (\gamma , d\frac{t-s}{2} )-2d \pi (\gamma ,s) )}_{ \tau } \end{aligned}$$

for all \(d\in \mathbb{D}^{1}\), \(t,s\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). Then

$$\begin{aligned}& \xi ^{\varTheta (\gamma ,d(t+s))-d \varTheta (\gamma ,t)-d \varTheta ( \gamma ,s)}_{\tau }\ast \xi ^{\pi (\gamma ,d(t+s))+\pi (\gamma ,d(t-s))-2d \pi (\gamma ,s)}_{\tau } \\& \quad \geq \xi ^{ \omega (2\varTheta (\gamma , d\frac{t+s}{2} )-d \varTheta (\gamma ,t)-d \varTheta (\gamma ,s) )}_{\tau }\ast \xi ^{ \nu (2\pi (\gamma ,d\frac{t+s}{2} )+2\pi ( \gamma , d\frac{t-s}{2} )-2d \pi (\gamma ,s) )}_{\tau } \end{aligned}$$
(3.13)

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

$$\begin{aligned}& \xi ^{\varTheta (\gamma ,d(t+s))-d \varTheta (\gamma ,t)-d \varTheta ( \gamma ,s)}_{\tau }\ast \xi ^{\pi (\gamma ,d(t+s))+\pi (\gamma ,d(t-s))-2d \pi (\gamma ,s)}_{\tau } \\& \quad \geq \xi ^{ \omega ( \varTheta (\gamma , d(t+s) )-d \varTheta (\gamma ,t)-d \varTheta (\gamma ,s) )}_{\tau } \ast \xi ^{ \nu (\pi (\gamma ,d(t+s) )+\pi ( \gamma , d(t-s) )-2d \pi (\gamma ,s) )}_{\tau } \end{aligned}$$
(3.14)

for all \(d\in \mathbb{D}^{1}\) and \(t,s\in T\), \(\gamma \in \varGamma \), \(\tau >0\), which implies that

$$\begin{aligned}& \varTheta \bigl(\gamma ,d(t+s)\bigr)-d \varTheta (\gamma ,t)-d \varTheta (\gamma ,s)=0, \\& \pi \bigl(\gamma ,d(t+s)\bigr)+\pi \bigl(\gamma ,d(t-s)\bigr)-2d \pi (\gamma ,s)=0. \end{aligned}$$

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

$$\begin{aligned}& \xi ^{[\varTheta ,\phi ] (\gamma ,ts)-[\varTheta ,\phi ](\gamma ,t)s-t[ \varTheta ,\phi ](\gamma ,s)}_{\tau }\ast \xi ^{\pi (\gamma ,ts)-\pi ( \gamma ,t)s-t \pi (\gamma ,s)}_{\tau } \\& \quad \geq \xi ^{[G,H] (\gamma ,\frac{ts}{4^{n}})-[G,H](\gamma , \frac{t}{2^{n}})\frac{s}{2^{n}}-\frac{t}{2^{n}}[G,H](\gamma , \frac{s}{2^{n}})}_{\frac{\tau }{4^{n}}}\ast \xi ^{H(\gamma , \frac{ts}{4^{n}})-H(\gamma ,\frac{t}{2^{n}})\frac{s}{2^{n}}- \frac{t}{2^{n}} H(\gamma ,\frac{s}{2^{n}})}_{\frac{\tau }{4^{n}}} \\& \quad \geq \varphi ^{\frac{t}{2^{n}},\frac{s}{2^{n}}}_{\frac{\tau }{4^{n}}} \geq \varphi ^{t,t}_{\frac{\tau }{\beta ^{n}}}, \end{aligned}$$
(3.15)

which tends to 1 as \(n\to +\infty \). Then

$$\begin{aligned}& [\varTheta ,\phi ] (\gamma ,ts)-[\varTheta ,\phi ](\gamma ,t)s-t[\varTheta , \phi ]( \gamma ,s)= 0, \\& \pi (\gamma ,ts)-\pi (\gamma ,t)s-t \pi (\gamma ,s)= 0, \end{aligned}$$

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

$$\begin{aligned}& \xi ^{G(\gamma ,d(t+s))-d G(\gamma ,t)-d G(\gamma ,s)}_{ \tau } \ast \xi ^{H(\gamma ,d(t+s))+H(\gamma ,d(t-s))-2d H(\gamma ,t)}_{ \tau } \\& \quad \geq \xi ^{ \omega (2G (\gamma , d\frac{t+s}{2} )-d G(\gamma ,t)-d G(\gamma ,s) )}_{\tau } \\& \qquad {} \ast \xi ^{ \nu (2H (\gamma ,d\frac{t+s}{2} )+2H (\gamma ,d\frac{t-s}{2} )-2d H(\gamma ,t) )}_{\tau }\ast \frac{\tau }{\tau +q( \Vert t \Vert ^{p}+ \Vert s \Vert ^{p})} \end{aligned}$$
(3.16)

for all\(d\in \mathbb{D}^{1}\), \(t,s\in T\), \(\gamma \in \varGamma \)and\(\tau >0\). Let

$$ \xi ^{[G,H] (\gamma ,ts)-[G,H](\gamma ,t)s-t[G,H](\gamma ,s)}_{\tau } \ast \xi ^{H(\gamma ,ts)-H(\gamma ,t)s-t H(\gamma ,s)}_{\tau }\geq \frac{\tau }{\tau +q( \Vert t \Vert ^{p}+ \Vert s \Vert ^{p})} $$
(3.17)

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

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau }\geq \frac{\tau }{\tau +q(\frac{2}{2^{p}-2} \Vert t \Vert ^{p})} $$
(3.18)

for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).

Proof

In Theorem 3.3, putting

$$ \varphi ^{t,s}_{\tau }=\frac{\tau }{\tau +q( \Vert t \Vert ^{p}+ \Vert s \Vert ^{p})} $$

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

$$ \varphi ^{t,s}_{4\beta \tau }\ge \varphi ^{\frac{t}{2},\frac{s}{2}}_{ \tau } $$
(3.19)

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

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau }\geq \varphi ^{t,t}_{2(1-\beta )\tau } $$
(3.20)

for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).

Proof

Using (3.6), we get

$$ \xi ^{G(\gamma ,t)-\frac{1}{2}G (\gamma ,2t )}_{\tau } \ast \xi ^{H(\gamma ,t)-\frac{1}{2}H (\gamma ,2t )}_{\tau } \geq \varphi ^{2t,2t}_{2\tau } \geq \varphi ^{t,t}_{ \frac{\tau }{2\beta }} $$
(3.21)

for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\).

Replacing t by \(2^{n}t\) in (3.21), we get

$$\begin{aligned} \xi ^{\frac{1}{2^{n}} G (\gamma ,2^{n}t )- \frac{1}{2^{n+1}}G (\gamma ,2^{n+1}t )}_{\tau }\ast \xi ^{ \frac{1}{2^{n}} H (\gamma ,2^{n}t )-\frac{1}{2^{n+1}}H (\gamma ,2^{n+1}t )}_{\tau } \geq & \varphi ^{2^{n+1}t,2^{n+1}t}_{2^{n+1} \tau } \\ \geq & \varphi ^{t,t}_{\frac{2^{n+1}}{(4\beta )^{n}}\tau } \end{aligned}$$
(3.22)

for all \(t\in T\), \(\gamma \in \varGamma \), \(\tau >0\) and \(n\in \mathbb{N}\). Since

$$ \frac{1}{2^{n}}G \bigl(\gamma ,{2^{n}t} \bigr)-G(\gamma ,t)= \sum_{k=0}^{n-1} \frac{1}{2^{k+1}}G \bigl(\gamma ,2^{k+1}t \bigr)-\frac{1}{2^{k}}G \bigl(\gamma ,2^{k}t \bigr), $$

we have

$$\begin{aligned}& \xi ^{\frac{1}{2^{n}} G (\gamma ,2^{n}t )-G ( \gamma ,t )}_{\sum _{k=0}^{n-1}\frac{(4\beta )^{k}}{2^{k+1}} \tau }\ast \xi ^{\frac{1}{2^{n}} H (\gamma ,2^{n}t )-H (\gamma ,t )}_{\sum _{k=0}^{n-1} \frac{(4\beta )^{k}}{2^{k+1}}\tau } \\& \quad \geq \prod_{k=0}^{n-1} \bigl[ \xi ^{\frac{1}{2^{k+1}}G ( \gamma ,2^{k+1}t )-\frac{1}{2^{k}}G (\gamma ,2^{k}t )}_{ \frac{(4\beta )^{k}}{2^{k+1}}\tau }\ast \xi ^{\frac{1}{2^{k+1}}H (\gamma ,2^{k+1}t )-\frac{1}{2^{k}}H (\gamma ,2^{k}t )}_{\frac{(4\beta )^{k}}{2^{k+1}}\tau } \bigr] \\& \quad \geq \varphi ^{t,t}_{\tau } \end{aligned}$$
(3.23)

and so

$$ \xi ^{\frac{1}{2^{n}} G (\gamma ,2^{n}t )-G (\gamma ,t )}_{\tau }\ast \xi ^{\frac{1}{2^{n}} H (\gamma ,2^{n}t )-H (\gamma ,t )}_{\tau } \geq \varphi ^{t,t}_{ \frac{\tau }{\sum _{k=0}^{n-1}\frac{(4\beta )^{k}}{2^{k+1}}}} $$
(3.24)

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

$$\begin{aligned} \xi ^{\frac{1}{2^{n+m}} G (\gamma ,2^{n+m}t )- \frac{1}{2^{m}}G (\gamma , 2^{m}t )}_{\tau }\ast \xi ^{ \frac{1}{2^{n+m}} H (\gamma ,2^{n+m}t )-\frac{1}{2^{m}}H (\gamma , 2^{m}t )}_{\tau } \geq & \varphi ^{2^{m}t,2^{m}t}_{ \frac{\frac{1}{2^{m}}\tau }{\sum _{k=0}^{n-1}\frac{(4\beta )^{k}}{2^{k+1}}}} \\ \geq & \varphi ^{t,t}_{ \frac{\tau }{\sum _{k=m}^{n+m}\frac{(4\beta )^{k}}{2^{k+1}}}} \end{aligned}$$
(3.25)

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

$$ \varTheta (\gamma ,t):=\lim_{n\rightarrow +\infty } \frac{1}{2^{n} }G \bigl(\gamma ,2^{n}t \bigr),\qquad \pi (\gamma ,t):= \lim_{n\rightarrow +\infty }\frac{1}{2^{n} }G \bigl(\gamma ,2^{n}t \bigr) , $$
(3.26)

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

$$\begin{aligned}& \xi ^{[\varTheta ,\phi ] (\gamma ,ts)-[\varTheta ,\phi ](\gamma ,t)s-t[ \varTheta ,\phi ](\gamma ,s)}_{\tau }\ast \xi ^{\pi (\gamma ,ts)-\pi ( \gamma ,t)s-t \pi (\gamma ,s)}_{\tau } \\& \quad \geq \xi ^{[G,H] (\gamma ,4^{n} ts)-[G,H](\gamma ,2^{n} t)2^{n} s-2^{n} t[G,H](\gamma ,2^{n} s)}_{4^{n}\tau }\ast \xi ^{H(\gamma ,4^{n} ts)-H( \gamma ,2^{n} t)2^{n} s-2^{n} t H(\gamma ,2^{n} s)}_{4^{n}\tau } \\& \quad \geq \varphi ^{2^{n}t,2^{n}s}_{4^{n}\tau } \\& \quad \geq \varphi ^{t,t}_{\frac{\tau }{\beta ^{n}}}, \end{aligned}$$
(3.27)

which tends to 1 as \(n\rightarrow +\infty \). Then

$$\begin{aligned}& [\varTheta ,\phi ] (\gamma ,ts)-[\varTheta ,\phi ](\gamma ,t)s-t[\varTheta , \phi ]( \gamma ,s)= 0, \\& \pi (\gamma ,ts)-\pi (\gamma ,t)s-t \pi (\gamma ,s)= 0 \end{aligned}$$

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

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau }\geq \frac{\tau }{\tau +q(\frac{2}{2-2^{p}} \Vert t \Vert ^{p})} $$
(3.28)

for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).

Proof

In Theorem 3.5, putting

$$ \varphi ^{t,s}_{\tau }=\frac{\tau }{\tau +q( \Vert t \Vert ^{p}+ \Vert s \Vert ^{p})}, $$

and letting \(\beta =2^{p-1}\), we get the desired result. □

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

$$ \varphi ^{\frac{t}{2},\frac{s}{2}}_{\frac{\beta }{2}\tau }\ge \varphi ^{\frac{t}{2},\frac{s}{2}}_{\frac{\beta }{4}\tau }\ge \varphi ^{t,s}_{\tau } $$
(4.1)

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

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau }\geq \varphi ^{t,t}_{ \frac{2(1-\beta )}{\beta }\tau } $$
(4.2)

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

$$ \xi ^{G(\gamma ,2t)-2G(\gamma ,t)}_{\tau }* \xi ^{H(\gamma ,2t)-2H( \gamma ,t)}_{\tau }\geq \varphi ^{t,t}_{\tau } $$
(4.3)

and so

$$\begin{aligned} \xi ^{G(\gamma ,t)-2G (\gamma ,\frac{t}{2} )}_{\tau }\ast \xi ^{H(\gamma ,t)-2H (\gamma ,\frac{t}{2} )}_{\tau } \geq & \varphi ^{\frac{t}{2},\frac{t}{2}}_{\tau } \\ \geq & \varphi ^{t,t}_{\frac{2}{\beta }\tau } \end{aligned}$$

for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\).

On the set

$$ S:=\bigl\{ (G,H)\mid G,H:\varGamma \times T \to T, G(\gamma ,0)=H(\gamma ,0)=0 \bigr\} , $$

we define the following generalized metric on S:

$$\begin{aligned}& \delta \bigl((G,H),(G_{1},H_{1})\bigr) \\& \quad =\inf \bigl\{ \mu \in \mathbb{R}_{+}:\xi ^{ G ( \gamma ,t)-G_{1}(\gamma ,t)}_{\tau }*\xi ^{ H(\gamma ,t)-H_{1}( \gamma ,t)}_{\tau }\geq \varphi ^{t,t}_{\frac{\tau }{\mu }}, \forall t \in T,\gamma \in \varGamma ,\tau >0\bigr\} . \end{aligned}$$

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

$$ \varLambda (G,H) (\gamma ,t):= \biggl(2G \biggl(\gamma ,\frac{t}{2} \biggr),2H \biggl(\gamma ,\frac{t}{2} \biggr) \biggr) $$

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

$$ \xi ^{ G (\gamma ,t)-G_{1}(\gamma ,t)}_{\tau }*\xi ^{ H(\gamma ,t)-H_{1}( \gamma ,t)}_{\tau } \geq \varphi ^{t,t}_{\frac{\tau }{\varepsilon }} $$

for all \(t\in T\), \(\gamma \in \varGamma \) and \(\tau >0\). So

$$ \xi ^{ 2G (\gamma ,\frac{t}{2})-2G_{1}(\gamma ,\frac{t}{2})}_{\tau }* \xi ^{2 H(\gamma ,\frac{t}{2})-H_{1}(\gamma ,\frac{t}{2})}_{\tau } \geq \varphi ^{\frac{t}{2},\frac{t}{2}}_{\frac{\tau }{\varepsilon }} \geq \varphi ^{t,t}_{\frac{\tau }{\beta \varepsilon }} $$

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

$$ \delta \bigl(\varLambda (G,H),\varLambda (G_{1},H_{1})\bigr) \leq \beta \delta \bigl((G,H),(G_{1},H_{1})\bigr) $$

for all \((G,H),(G_{1},H_{1})\in S\).

It follows from (3.3) that

$$ \xi ^{ G (\gamma ,t)-2G_{1}(\gamma ,\frac{t}{2})}_{\tau }*\xi ^{ H( \gamma ,t)-H_{1}(\gamma ,\frac{t}{2})}_{\tau } \geq \varphi ^{ \frac{t}{2},\frac{t}{2}}_{\tau }\geq \varphi ^{t,t}_{ \frac{2\tau }{\beta }} $$

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

$$ \varTheta (\gamma ,t):=2 \varTheta \biggl(\gamma , \frac{t}{2} \biggr), \qquad \pi (\gamma ,t):=2 \pi \biggl(\gamma , \frac{t}{2} \biggr) $$
(4.4)

for all \(t\in T\), \(\gamma \in \varGamma \). The random operator \((\varTheta ,\pi )\) is a unique fixed point of Λ in the set

$$ M=\bigl\{ (G,H)\in S : \delta \bigl((G,H),(G_{1},H_{1}) \bigr)< \infty \bigr\} . $$

(2) \(\delta (\varLambda ^{n}(G,H),(\varTheta ,\pi ))\to 0\) as \(n\rightarrow +\infty \). which implies

$$ \varTheta (\gamma ,t):=\lim_{n\to +\infty }2^{n} G \biggl( \gamma , \frac{t}{2^{n}} \biggr), \qquad \pi (\gamma ,t):=\lim_{n \to +\infty }2^{n} H \biggl(\gamma ,\frac{t}{2^{n}} \biggr). $$

(3) \(\delta ((G,H),(\varTheta ,\pi ))\leq \frac{1}{1-\beta }\delta ((G,H), \varLambda (G,H))\), which implies

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau } \geq \varphi ^{t,t}_{ \frac{2(1-\beta )}{\beta }\tau } $$

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

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau }\geq \exp \biggl(- \frac{q(\frac{2}{2^{p}-2} \Vert t \Vert ^{p})}{\tau } \biggr) $$

for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).

Proof

In Theorem 4.1, putting

$$ \varphi ^{t,s}_{\tau }=\exp \biggl(- \frac{q(\frac{2}{2^{p}-2} \Vert t \Vert ^{p})}{\tau } \biggr), $$

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

$$ \varphi ^{t,s}_{4\beta \tau }\ge \varphi ^{\frac{t}{2},\frac{s}{2}}_{ \tau } $$
(4.5)

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

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau }\geq \varphi ^{t,t}_{2(1-\beta )\tau } $$
(4.6)

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

$$ \varLambda (G,H) (\gamma ,t):= \biggl(\frac{1}{2}G(\gamma ,2t), \frac{1}{2}H( \gamma ,2t) \biggr) $$

for all \(t\in T\), \(\gamma \in \varGamma \). It follows from (4.3) that

$$\begin{aligned} \xi ^{G(\gamma ,t)-\frac{1}{2}G (\gamma ,2t )}_{\tau } \ast \xi ^{H(\gamma ,t)-\frac{1}{2}H (\gamma ,2t )}_{\tau } \geq & \varphi ^{2t,2t}_{2\tau } \\ \geq & \varphi ^{t,t}_{\frac{\tau }{2\beta }} \end{aligned}$$

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

$$ \xi ^{G(\gamma ,t)-\varTheta (\gamma ,t)}_{\tau }\ast \xi ^{H(\gamma ,t)- \pi (\gamma ,t)}_{\tau }\geq \exp \biggl(- \frac{q(\frac{2}{2-2^{p}} \Vert t \Vert ^{p})}{\tau } \biggr) $$

for all\(t\in T\), \(\gamma \in \varGamma \)and\(\tau >0\).

Proof

In Theorem 4.3, putting

$$ \varphi ^{t,s}_{\tau }=\exp \biggl(- \frac{q(\frac{2}{2-2^{p}} \Vert t \Vert ^{p})}{\tau } \biggr), $$

and letting \(\beta =2^{p-1}\), we get the desired result. □

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Acknowledgements

The authors are thankful to the anonymous referees for giving valuable comments and suggestions which helped to improve the final version of this paper.

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Funding

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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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.

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Correspondence to Reza Saadati.

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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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MSC

  • 47B47
  • 47H10
  • 39B52
  • 39B72
  • 46L57

Keywords

  • Lie bracket (derivation, derivation)
  • Stability
  • Stochastic controller
  • Fixed point technique
  • Banach algebra
  • Random operator inequality
  • Menger space