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.

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

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

(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).

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:

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

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 [1–3]).

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

• (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

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

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

• (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 [11–30].

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

Theorem 2.5

([31–33])

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) $\delta (\varLambda ^{n}t,\varLambda ^{n+1}t)<\infty$, 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

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

(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. □

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

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

(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. □