Skip to main content

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. □

References

  1. 1.

    Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland, New York (1983)

    Google Scholar 

  2. 2.

    S̆erstnev, A.N.: On the concept of a stochastic normalized space. Dokl. Akad. Nauk SSSR 149, 280–283 (1963) (in Russian)

    MathSciNet  Google Scholar 

  3. 3.

    Saadati, R.: Random Operator Theory. Elsevier/Academic Press, London (2016)

    Google Scholar 

  4. 4.

    Hadzic, O., Pap, E. (eds.): Mathematics and Its Applications, vol. 536. Kluwer Academic, Dordrecht (2001)

    Google Scholar 

  5. 5.

    Mirmostafaee, A.K.: Perturbation of generalized derivations in fuzzy Menger normed algebras. Fuzzy Sets Syst. 195, 109–117 (2012)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Saadati, R., Park, C.: Approximation of derivations and the superstability in random Banach -algebras. Adv. Differ. Equ. 2018, Paper No. 418 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Naeem, R., Anwar, M.: Jessen type functionals and exponential convexity. J. Math. Comput. Sci. 17, 429–436 (2017)

    Article  Google Scholar 

  8. 8.

    Park, C., Yun, S.: Stability of cubic and quartic ρ-functional inequalities in fuzzy normed spaces. J. Nonlinear Sci. Appl. 9, 1693–1701 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Maleki, M.V., Vaezpour, S.M., Saadati, R.: Nonlinear stability of ρ-functional equations in latticetic random Banach lattice spaces. Mathematics 6(2), Paper No. 22 (2018)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Agarwal, R.P., Saadati, R., Salamati, A.: Approximation of the multiplicatives on random multi-normed space. J. Inequal. Appl. 2017, Paper No. 204 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Jang, S.Y., Saadati, R.: Approximation of an additive \((\varrho _{1},\varrho _{2})\)-random operator inequality. J. Funct. Spaces 2020, Article ID 7540303 (2020)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Park, C., Eshaghi Gordji, M., Saadati, R.: Random homomorphisms and random derivations in random normed algebras via fixed point method. J. Inequal. Appl. 2012, Paper No. 194 (2012)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Rassias, J.M., Saadati, R., Sadeghi, G., Vahidi, J.: On nonlinear stability in various random normed spaces. J. Inequal. Appl. 2011, Paper No. 62 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Cho, Y.J., Rassias, T.M., Saadati, R.: Stability of Functional Equations in Random Normed Spaces, vol. 86. Springer, New York (2013)

    Google Scholar 

  15. 15.

    Lu, G., Xin, J., Jin, Y., Park, C.: Approximation of general Pexider functional inequalities in fuzzy Banach spaces. J. Nonlinear Sci. Appl. 12, 206–216 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    El-Moneam, M.A., Ibrahim, T.F., Elamody, S.: Stability of a fractional difference equation of high order. J. Nonlinear Sci. Appl. 12, 65–74 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Keltouma, B., Elhoucien, E., Rassias, T.M., Ahmed, R.: Superstability of Kannappan’s and Van Vleck’s functional equations. J. Nonlinear Sci. Appl. 11, 894–915 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Ding, Y.: Ulam–Hyers stability of fractional impulsive differential equations. J. Nonlinear Sci. Appl. 11, 953–959 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Binzar, T., Pater, F., Nadaban, S.: On fuzzy normed algebras. J. Nonlinear Sci. Appl. 9, 5488–5496 (2016)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hadz̆ić, O.: A random fixed point theorem for multivalued mappings of Ćirić’s type. Mat. Vesn. 3(16)(31)(4), 397–401 (1979)

    Google Scholar 

  21. 21.

    Todorc̆ević, V.: Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics. Springer, Cham (2019)

    Google Scholar 

  22. 22.

    Patle, P., Patel, D., Aydi, H., Radenović, S.: On \(\mathcal{H}^{+}\) type multivalued contraction and its applications in symmetric and probabilistic spaces. Mathematics 7(2), Paper No. 144 (2019)

    Article  Google Scholar 

  23. 23.

    Ndolane, S.: Exponential form for Lyapunov function and stability analysis of the fractional differential equations. J. Math. Comput. Sci. 18, 388–397 (2018)

    Article  Google Scholar 

  24. 24.

    Wu, R., Li, L.: Note on the stability property of the boundary equilibrium of a May cooperative system with strong and weak cooperative partners. J. Math. Comput. Sci. 20, 58–63 (2020)

    Article  Google Scholar 

  25. 25.

    Lee, Y., Jung, S.: A fixed point approach to the stability of a general quartic functional equation. J. Math. Comput. Sci. 20, 207–215 (2020)

    Article  Google Scholar 

  26. 26.

    Piri, H., Rahrovi, S., Kumam, P.: Generalization of Khan fixed point theorem. J. Math. Comput. Sci. 17, 76–83 (2017)

    Article  Google Scholar 

  27. 27.

    Shoaib, A., Azam, A., Arshad, M., Ameer, E.: Fixed point results for multivalued mapping on sequence in a closed ball with applications. J. Math. Comput. Sci. 17, 308–316 (2017)

    Article  Google Scholar 

  28. 28.

    Brzdek, J., Ciepliński, K.: A fixed point theorem in n-Banach spaces and Ulam stability. J. Math. Anal. Appl. 470, 632–646 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Park, C.: Lie bracket derivation-derivations in complex Banach algebras. Preprint

  30. 30.

    Nădăban, S., Bînzar, T., Pater, F.: Some fixed point theorems for φ-contractive mappings in fuzzy normed linear spaces. J. Nonlinear Sci. Appl. 10(11), 5668–5676 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4(1), Article 4 (2003)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Cădariu, L., Radu, V.: Fixed points and the stability of quadratic functional equations. An. Univ. Vest. Timiş., Ser. Mat.-Inform. 41, 25–48 (2003)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Park, C.: Homomorphisms between Poisson \(JC^{\ast }\)-algebras. Bull. Braz. Math. Soc. 36, 79–97 (2005)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Miheţ, D., Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Miheţ, D., Saadati, R.: On the stability of the additive Cauchy functional equation in random normed spaces. Appl. Math. Lett. 24, 2005–2009 (2011)

    MathSciNet  Article  Google Scholar 

Download references

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.

Availability of data and materials

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).

Author information

Affiliations

Authors

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.

Corresponding author

Correspondence to Reza Saadati.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

MSC

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

Keywords

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