# 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 ).

### 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

()

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 , fuzzy metric spaces , fuzzy normed spaces , non-Archimedian random Lie $$C^{*}$$-algebras , random multi-normed space , non-Archimedean random normed spaces ; see also .

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

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

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 , Miheţ and Radu proved that $$(S, \delta )$$ is complete (see also ).

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.

Not applicable.

## 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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### Contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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### Competing interests

The authors declare that they have no competing interests.

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