Hypolimnetic assimilation of ammonium by the nuisance alga <em>Gonyostomum semen</em>

Thomas Rohrlack; Thomas Rohrlack

doi:10.3934/microbiol.2020006

AIMS Microbiology

2020, Volume 6, Issue 2: 92-105. doi: 10.3934/microbiol.2020006

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Research article

Hypolimnetic assimilation of ammonium by the nuisance alga Gonyostomum semen

Thomas Rohrlack ^,

Norwegian University of Life Sciences, Faculty for Environmental Sciences and Natural Resource Management, Postbox 5003, NO-1432 Ås, Norway

Received: 27 February 2020 Accepted: 17 April 2020 Published: 22 April 2020

Gonyostomum semen is a bloom-forming freshwater raphidophyte that is currently on the increase, which concerns water managers and ecologists alike. Much indicates that the recent success of G. semen is linked to its diel vertical migration (DVM), which helps to overcome the spatial separation of optimal light conditions for photosynthesis at the surface of a lake and the high concentration of phosphate in the hypolimnion. I here present data from a field study conducted in Lake Lundebyvannet (Norway) in 2017–2019 that are consistent with the idea that the DVM of G. semen also allows for a hypolimnetic uptake of ammonium. As expected, microbial mineralization of organic matter in a low-oxygen environment led to an accumulation of ammonium in the hypolimnion as long as G. semen was absent. In contrast, a decreasing or constantly lower concentration of hypolimnetic ammonium was found in presence of a migrating G. semen population. In summer of 2019, a short break in the DVM of G. semen coincided with a rapid accumulation of hypolimnetic ammonium, which was equally rapidly decimated when G. semen resumed its DVM. Taken together, these data support the idea that G. semen can exploit the hypolimnetic pool of ammonium, which may be one reason for the recent success of the species and its significant impact on the structure of the aquatic food web.

Keywords:

Citation: Thomas Rohrlack. Hypolimnetic assimilation of ammonium by the nuisance alga Gonyostomum semen[J]. AIMS Microbiology, 2020, 6(2): 92-105. doi: 10.3934/microbiol.2020006

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Abstract

1. Introduction

Hyers ^[1] made a response to the question of Ulam in the context of Banach spaces in relation to additive mappings and was a considerable step towards further solutions in this area. Note the concept of stability is a major property in the qualitative theory of differential equations. Over the last few years, results have been presented on numerous types of differential equations. The approach proposed by Hyers ^[1] which provides the additive function is named a direct technique. This technique is a significant and helpful tool used to investigate the stability of different functional equations. In recent years, a number of research monographs and articles have been studied on diverse applications and generalizations of the HUS, like k-additive mappings, differential equations, Navier–Stokes equations, ODEs, and PDEs (see ^[2,3,4]). Also in recent years, the stability of different (integral and differential, others functional) equations and other subjects (such as $C^{*}$ -ternary algebras, groups, flows and Banach algebras) have been investigated. Fixed–point methods are useful when examining stability and fixed point theory proposes vital tools for solving problems arising in different fields of functional analysis, like equilibrium problems, differential equations, and dynamical systems.

Assume Banach algebras $\mathscr{Q}$ and $\mathscr{Q}^{\prime\prime}$ . Suppose $(\mathscr{Q}^{\prime}, \Delta)$ is a probability measure space and suppose $(\mathscr{Q}, {\mathfrak B}_{\mathscr{Q}})$ and $(\mathscr{Q}^{\prime\prime}, \mathfrak{B}_{\mathscr{Q}^{\prime\prime}})$ are Borel measurable spaces. Then a map $f:\mathscr{Q}^{\prime}\times\mathscr{Q}\to \mathscr{Q}^{\prime\prime}$ is a operator if $\{\wp:\; f(\wp, \alpha)\in \nu\}\in\Delta$ for each $\alpha$ in $\mathscr{Q}$ and $\nu\in \mathfrak{B}_{\mathscr{Q}^{\prime\prime}}$ . Assume $\mho = (\mho_{1}, \ldots, \mho_{m})$ and $\Omega = (\Omega_{1}, \ldots, \Omega_{m}), m\in \mathbb{N}$ . Then we have

$\mho \preceq \Omega \Longleftrightarrow \mho_{\imath}\leq\Omega_{\imath}, \:\:\:\:\imath = 1,\cdots,m;$

and also

$\mho \rightarrow 0 \Longleftrightarrow \mho_{\imath}\rightarrow 0, \:\:\:\:\imath = 1,\cdots,m.$

Definition 1.1 (^[5]). Let $\nabla\neq\emptyset$ is a set and $d: \nabla^{2}\rightarrow [0, +\infty]^{m}, m\in \mathbb{N},$ is a given mapping. If the following conditions are satisfied, then we say $d$ is a generalized metric on $\nabla$ :

(1) For each $(g, g^{\prime})\in \nabla \times \nabla$ , we get

$d(g,g^{\prime}) = \underbrace{(0,\cdots , 0)}_{m} \Longleftrightarrow \: g = g^{\prime};$

(2) For each $(g, g^{\prime})\in \nabla\times \nabla$ , we get

$d(g^{\prime},g) = d(g,g^{\prime}) \Longleftrightarrow \: g = g^{\prime};$

(3) For each $g, g^{\prime}, g^{\prime\prime}\in \nabla$ , we get

$d(g,g^{\prime\prime})+d(g^{\prime\prime},g^{\prime}) \succeq d(g^{\prime},g).$

Theorem 1.2 (^[5]). Assume the following assumptions:

(1) $d: \nabla^{2}\rightarrow [0, +\infty]^{m}, m\in \mathbb{N},$ and $(\nabla, d)$ is a complete generalized metric space.

(2) $\mathscr{L}: \nabla \rightarrow \nabla$ is a strictly contractive mappingwith Lipschitz constant $\mathcal{Z} < 1$ .

Then for each $g\in \nabla$ , either

$d(\mathscr{L}^n g,\mathscr{L}^{n+1} g) = \overbrace{(+\infty, \cdots , +\infty)}^{m}$

for each $n\in \mathbb{N}\cup \{ 0\}$ or there is a $n_0\in \mathbb{N}$ such that

(1) $d(\mathscr{L}^n g, \mathscr{L}^{n+1}g) \preceq \overbrace{(+\infty, \cdots, +\infty)}^{m}, \qquad \forall n\ge n_0;$

(2) The sequence $\{\mathscr{L}^n g\}$ converges to a fixed point $(g^{\prime})^*$ of $\mathscr{L}$ ;

(3) $(g^{\prime})^*$ is the unique fixed point of $\mathscr{L}$ in the set $\complement = \{g^{\prime}\in \nabla\mid d(\mathscr{L}^{n_0} g, g^{\prime}) \preceq \overbrace{(+\infty, \cdots, +\infty)}^{m}\}$ ;

(4) $d(g^{\prime}, (g^{\prime})^*) \preceq \frac{1}{1-\mathcal{Z} } d(g^{\prime}, \mathscr{L}g^{\prime})$ for each $g^{\prime} \in \complement$ .

We use fixed-point way to study the multi-stability of antiderivations associated with the following inequality:

$\begin{align} &\text{diag}\bigg[\:\: \bigg\Vert f_{1}( \Lambda,\varepsilon+\zeta+\varsigma)-f_{1}(\Lambda ,\varepsilon+\varsigma)-f_{1}(\Lambda ,\varepsilon+\zeta-\varsigma)+f_{1}(\Lambda ,\varepsilon-\varsigma))\bigg\Vert ,\ldots , \\ &\:\:\:\:\:\:\:\:\:\:\:\:\:\:\bigg\Vert f_{n}( \Lambda ,\varepsilon+\zeta+\varsigma)-f_{n}(\Lambda ,\varepsilon+\varsigma)-f_{n}(\Lambda ,\varepsilon+\zeta-\varsigma)+f_{n}(\Lambda ,\varepsilon-\varsigma))\bigg\Vert \:\:\bigg]_{n\times n}\\ &\preceq \text{diag}\bigg[ \:\: \bigg\Vert \theta_{1}\bigg(f_{1}(\Lambda ,\varepsilon-\zeta+\varsigma)-f_{1}(\Lambda ,\varepsilon+\varsigma)-f_{1}(\Lambda ,\varepsilon-\zeta-\varsigma)+f_{1}(\Lambda ,\varepsilon-\varsigma)\bigg)\bigg\Vert,\ldots ,\\ &\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\bigg\Vert \theta_{n}\bigg(f_{n}(\Lambda ,\varepsilon-\zeta+\varsigma)-f_{n}(\Lambda ,\varepsilon+\varsigma)-f_{n}(\Lambda ,\varepsilon-\zeta-\varsigma)+f_{n}(\Lambda ,\varepsilon-\varsigma)\bigg)\bigg\Vert\:\:\bigg]_{n\times n} \end{align}$

(1.1)

for each $\varepsilon, \zeta, \varsigma \in \mathscr{Q}$ , $\Lambda\in \mathscr{Q}^{\prime}$ with $\vert \theta_{1} \vert, \ldots, \vert \theta_{n} \vert < 1$ .

2. Proposing new control functions and the concept of multi stability

For this section we refer the reader ^[6,7]. Assume $\Re(\nu)$ denotes the real part of $\nu$ if $\nu\in \mathbb{C}$ . Also, let

(1) $\mathbb{Z}^{+}$ be the set of the positive integers;

(2) $\mathbb{Z}^{-}$ be the negative integer numbers;

(3) $\mathbb{R}_{-}$ be the negative real numbers;

(4) $\mathbb{R}_{+}$ be the positive real numbers.

We begin by defining various functions which will be needed later. The gamma function is given by

$\begin{eqnarray*} \Gamma(X) = \int_{0}^{\infty} e^{-Y}Y^{X-1}dY, \:\:\:\: \Re(X) > 0 ,\:\:\: X\in\mathbb{C}. \end{eqnarray*}$

Euler's functional equation is given by

$\Gamma (X+1) = X\Gamma (X), \:\:\: \Re (X) > 0,\: X\in \mathbb{C}.$

Theorem 2.1 (^[6]).If $X\in \mathbb{N}\cup \{0\},$ then

$\Gamma (X+1) = X!.$

Theorem 2.2 (^[6]). $\Gamma(0.5) = \pi^{0.5}.$

The Pochhammer symbol is

$\begin{eqnarray*} (\complement)_{\jmath} = \prod^{\jmath}_{\imath = 1}(\complement + \imath-1) = \dfrac{\Gamma (\complement +\jmath)}{\Gamma(\complement)} = \left \{{\begin{array} {*{20}{c}} {1}\; \; \; \; \; \; \; \; \; \; \; \; \; {\jmath = 0}\\ {\complement (\complement +1) \cdots (\complement+\jmath -1)} {\jmath\in \mathbb{N}\cup\{ 0\}} \end{array}} \right. \end{eqnarray*}$

where $\complement\in \mathbb{C}$ and $\jmath, \imath\in\mathbb{N}.$

Note that

$\begin{eqnarray*} \Gamma(\complement+\jmath) = \complement (\complement +1)\cdots (\complement +\jmath-1)\Gamma (\complement) \end{eqnarray*}$

where $\jmath\in \mathbb{N}\cup \{0\}.$

The Gauss hypergeometric series ^[7] is given by

$\begin{eqnarray} && {\varphi_{1}^{{\circledS}}}(X): = {}_{2}\mathbb{F}_{1}(\alpha,\mathit{B};\mathit{T};X) \\ & = & 1+\dfrac{\alpha\mathit{B}}{\mathit{T}}X+\dfrac{\alpha(\alpha+1)\mathit{B}(\mathit{B}+1)}{\mathit{T}(\mathit{T}+1)}\dfrac{X^{2}}{2}+\cdots \\ & = &\sum\limits_{n = 0}^{\infty}\dfrac{(\alpha)_{n}(\mathit{B})_{n}}{(\mathit{T})_{n}}\dfrac{X^{n}}{n!}, \end{eqnarray}$

(2.1)

where $\alpha, \mathit{B}, \mathit{T}, X\in \mathbb{C}$ , $n\in\mathbb{N}\cup \{0\},$ and $\vert X\vert < 1.$

Consider the Gauss differential equation

$\begin{eqnarray} (X-X^{2})\dfrac{d^{2}\omega}{dX^{2}}+(\mathit{T}-(\alpha+\mathit{B}+1)X)\dfrac{d\omega}{dX}-\alpha\mathit{B}\omega = 0, \end{eqnarray}$

(2.2)

where $\alpha, \mathit{B}, X\in \mathbb{C}$ , $\mathit{T}\in \mathbb{C}\backslash (\mathbb{Z}^{-}\cup\{ 0\}),$ and $\vert X\vert < 1.$ The hypergeometric series is a solution of (2.2).

Theorem 2.3 (^[6]).Let $\alpha, \mathit{B}, \mathit{T}, X\in \mathbb{C}$ and $\vert X\vert < 1$ . Then

$\begin{eqnarray*} {}_{2}\mathbb{F}_{1}(\alpha,\mathit{B};\mathit{T};X) = \dfrac{\Gamma(\mathit{T})}{\Gamma(\mathit{B})\Gamma(\mathit{T}-\mathit{B})}\int_{0}^{1} Y^{\mathit{B} -1}(1-Y)^{\mathit{T}-\mathit{B}-1}(1-XY)^{-\alpha}dY, \end{eqnarray*}$

where $\Re(\mathit{T}) > \Re (\mathit{B}) > 0.$

Theorem 2.4. If $\Re(\mathit{T}) > 0, \vert X\vert < 1,$ and $\vert\arg (-X) \vert < \pi,$ then

$\begin{eqnarray*} {}_{2}\mathbb{F}_{1}(\alpha,\mathit{B};\mathit{T};X) = \dfrac{\Gamma(\mathit{T})}{\Gamma(\alpha)\Gamma(\mathit{B})}\dfrac{1}{2\pi i}\int_{-i\infty}^{+i\infty} \dfrac{\Gamma(\alpha+Y)\Gamma(\mathit{B}+Y)\Gamma(-Y)}{\Gamma(\mathit{T+Y})} (-X)^{Y}dY, \end{eqnarray*}$

We now present the Clausen hypergeometric series ^[7] and its properties:

$\begin{eqnarray} {\varphi_{2}^{{\circledS}}}(X)&: = & {}_{p}\mathbb{F}_{q}((\alpha);(\mathit{T}); X)\\ & = & {}_{p}\mathbb{F}_{q}(\alpha,\cdots , \alpha_{p};\mathit{T}_{1},\cdots , \mathit{T}_{q}; X)\\ & = & {}_{p}\mathbb{F}_{q}\bigg({}^{\alpha_{1},\cdots , \alpha_{p}}_{\mathit{T}_{1},\cdots , \mathit{T}_{q}} ; X\bigg)\\ & = & \sum\limits_{k = 0}^{\infty}\dfrac{(\alpha_{1})_{k}\cdots (\alpha_{p})_{k}}{(\mathit{T}_{1})_{k}\cdots (\mathit{T}_{q})_{k}}\dfrac{X^{n}}{n!}, \end{eqnarray}$

(2.3)

where $p, n, q\in \mathbb{N}\cup \{0\}$ and $\alpha_{n}, X, \mathit{T}_{n}\in\mathbb{C}.$

Now, (2.3) is a solution of the following differential equation

$\bigg(M(q,\eta , \mathit{T}_{n} ) \omega \bigg) (X) -\bigg(N(p,\eta , \alpha_{n} ) \omega \bigg) (X) = 0,$

where

$\begin{eqnarray*} &&\bigg(M(q,\eta , \mathit{T}_{n} ) \omega \bigg) (X)\\ & = & \bigg( X\dfrac{d}{dX} \bigg)\prod\limits_{n = 1}^{q}\bigg(\bigg( X\dfrac{d}{dX} \bigg) \omega (X)+(\mathit{T}_{n}-1) \omega (X)\bigg) = X\dfrac{d}{dX} \bigg(\prod\limits_{n = 1}^{q}\bigg(\bigg( X\dfrac{d}{dX} +(\mathit{T}_{n}-1)\bigg) \omega\bigg) (X)\bigg), \end{eqnarray*}$

and

$\begin{eqnarray*} \bigg(N(p,\eta , \alpha_{n} ) \omega \bigg) (X) = X\prod\limits_{n = 1}^{p}\bigg( X\dfrac{d\omega(X)}{dX} + \alpha_{n} \omega(X) \bigg) = X\prod\limits_{n = 1}^{p}\bigg(\bigg( X\dfrac{d}{dX} + \alpha_{n}\bigg) \omega \bigg)(X) \\ \end{eqnarray*}$

and $\alpha_{n}, X, \mathit{T}_{n}\in\mathbb{C},$ $p, n, q\in \mathbb{N}\cup \{0\},$ and $\vert X \vert < 1,$

Theorem 2.5 (^[6]).Suppose $\alpha_{n}\in\mathbb{C}\backslash (\mathbb{Z}^{-}\cup \{0\})$ :

(1) The series converges only for $X = 0,$ if $p > q+1.$

(2) The series converges absolutely for $X\in \mathbb{C},$ if $p < q.$

(3) The series converges absolutely for $\vert X \vert < 1$ and diverges for $\vert X \vert = 1$ and for $\vert X \vert > 1$ it converges absolutely for $\Re \bigg(\sum_{k = 1}^{q}\mathit{T}_{k} -\sum_{k = 1}^{p}\alpha_{k} \bigg) > 0,$ if $p = q+1.$

Assume the following notation ^[7]:

$\begin{eqnarray} \Xi : = -\sum\limits_{k = 1}^{q}b_{k}+\sum\limits_{j = 1}^{p}a_{j}, \end{eqnarray}$

(2.4)

$\begin{eqnarray} \sigma : = -\prod\limits_{k = 1}^{q}\vert b_{k}\vert^{- b_{k}}+\prod\limits_{j = 1}^{p}\vert a_{j}\vert^{- a_{j}}, \end{eqnarray}$

(2.5)

and

$\begin{eqnarray} \chi : = -\sum\limits_{j = 1}^{p}\kappa_{j}+\sum\limits_{k = 1}^{q}\vartheta_{k}+\dfrac{p-q}{2}, \end{eqnarray}$

(2.6)

where $\kappa_{j}, \vartheta_{k}\in\mathbb{C}, \: k, j\in \mathbb{N}, \:\: p, q\in\mathbb{N}\cup\{ 0\},$ and $b_{k}, a_{j} \in \mathbb{R}_{+}$ .

The Wright generalized hypergeometric series is given by

$\begin{eqnarray} {\varphi_{3}^{{\circledS}}}(X)&: = &{}_{p} \mathbb{W}_{q} (X)\\ & = & {}_{p} \mathbb{W}_{q} \bigg({}_{(\vartheta_{1},b_{1}),\cdots ,(\vartheta_{q},b_{q})}^{(\kappa_{1},a_{1}),\cdots ,(\kappa_{p},a_{p}) }; X\bigg)\\ & = & {}_{p} \mathbb{W}_{q} \bigg({}_{(\vartheta_{q},b_{q})_{1,q}}^{(\kappa_{p},a_{p})_{1,p} }; X\bigg)\\ & = & \sum\limits_{s = 0}^{\infty}\dfrac{\bigg\lbrace \prod_{j = 1}^{p}\Gamma (\kappa_{j}+a_{j}s)\bigg\rbrace}{\bigg\lbrace \prod_{k = 1}^{q}\Gamma (\vartheta_{k}+b_{k}s)\bigg\rbrace}\dfrac{X^{s}}{s!}, \end{eqnarray}$

(2.7)

where $j, s, k\in \mathbb{N}, X\in \mathbb{C}, \: \Xi > -1, \kappa_{j}, \vartheta_{k}\in \mathbb{C}, \:\: p, q\in\mathbb{N}\cup \{0\},$ and $b_{k}, a_{j}\in \mathbb{R}_{+}.$

Theorem 2.6 (^[6]).Suppose $X\in \mathbb{C}, \: \vartheta_{k}, \kappa_{j}\in \mathbb{C}, \: j, s, k\in \mathbb{N}, \:$ $b_{k}, a_{j}\in \mathbb{R}_{+},$ then

(1) (2.7) is absolutely convergent for each valueof $\vert X\vert = \sigma$ and of $\vert X\vert < \sigma,$ and $\Re (\chi) > 0.5,$ if $\Xi+1 = 0.$

(2) (2.7) is absolutely convergent for $X\in \mathbb{C},$ if $\Xi+1 > 0.$

Now, the Wright function is given by

$\begin{eqnarray} {\varphi_{4}^{{\circledS}}}(X): = \mathbb{K}(\vartheta , b , X) = {}_{0}\mathbb{W}_{1}\bigg( {}^{\:\:\:-}_{(b,\vartheta)} ; X \bigg) = \sum\limits_{k = 0}^{\infty} \dfrac{1}{\Gamma (\vartheta +bk)}\dfrac{X^{k}}{k!}, \end{eqnarray}$

(2.8)

where $X, \vartheta\in \mathbb{C},$ and $b\in \mathbb{R}.$

Theorem 2.7 (^[6]).Now (2.8) for $b\in \mathbb{C}$ $(b\in\mathbb{Z}^{-}\cup \{0\}$ if $\vartheta = 0)$ and $\vartheta > -1$ is an entire function of type $\delta = (1+\vartheta) \vert \vartheta\vert^{-\frac{\vartheta}{1+\vartheta}}$ , andfinite order $p = \frac{1}{1+\vartheta}.$

Theorem 2.8 (^[6]).Now (2.8) is an entire functionof $X$ for each $b\in \mathbb{C}$ and $\vartheta > -1.$

The Wright generalized Bessel function (Bessel-Maitland function) is given by

$\begin{eqnarray*} {\varphi_{5}^{{\circledS}}}(X): = \mathbb{J}(\kappa , a , X) = \sum\limits_{k = 0}^{\infty}\dfrac{1}{\Gamma (\kappa +1 +ak)}\dfrac{(-X)^{k}}{k!} = {}_{0}{\mathbb{W}}_{1}\bigg( {}^{\:\:\: -}_{(\kappa +1, b)} ;-X \bigg), \end{eqnarray*}$

where $\kappa, X\in \mathbb{C},$ and $a\in \mathbb{R}.$

Theorem 2.9 (^[6]).Suppose $X\in \mathbb{C}, \; j, s, k \in \mathbb{N},$ $a_{j}, b_{k}\in \mathbb{R}_{+},$ and $\kappa_{j}, \vartheta_{k}\in \mathbb{C}.$ Then (2.7) is an entire function of $X$ .

Theorem 2.10 (^[6]).Suppose $b\in \mathbb{R}$ and $\vartheta\in \mathbb{C}.$

(1) (2.8) is absolutely convergent for all $\vert X\vert < 1$ and of $\vert X\vert = 1$ , and $\Re (\chi) > 0.5,$ if $b+1 = 0.$

(2) (2.8) is absolutely convergent for $X\in \mathbb{C}$ , if $b+1 > 0.$

Theorem 2.11 (^[6]).Suppose $b > -1, \; \vartheta\in \mathbb{C}.$ Then(2.8) is anentire function of $X$ .

Theorem 2.12 (^[6]).Suppose $X\in \mathbb{C}, \: j, k, s\in \mathbb{N}, \: \kappa_{j}, \vartheta_{k}\in \mathbb{C},$ and $a_{j}, b_{k}\in \mathbb{R}_{+}.$ Then

$\begin{eqnarray*} {}_{p}\mathbb{W}_{q}\bigg( {}^{(\kappa_{1},1),\cdots , (\kappa_{p},1)}_{(\vartheta_{1},1),\cdots , (\vartheta_{q},1)} ; X \bigg) = \dfrac{\prod_{j = 1}^{p}\Gamma(\kappa_{j})}{\prod_{k = 1}^{p}\Gamma(\vartheta_{k})}{}_{p}\mathbb{F}_{q}\bigg( {}^{\kappa_{1},\cdots , \kappa_{p}}_{\vartheta_{1},\cdots , \vartheta_{q}}; X\bigg), \end{eqnarray*}$

where $\Xi+1\geq 0.$

The shifted Wright generalized hypergeometric series ^[6] is given by

$\begin{eqnarray*} && {\varphi_{6}^{{\circledS}}}(X)\\ &&: = {}_{p}{\mathbb{B}}_{q}(X)\\ && = {}_{p}{\mathbb{B}}_{q}\bigg( {}^{(\kappa_{1}, a_{1}; \vartheta_{1}, b_{1}), \cdots , (\kappa_{p}, a_{p}; \vartheta_{p}, b_{p})}_{(\widehat{\kappa}_{1}, c_{1}; \widehat{\vartheta}_{1}, d_{1}), \cdots , (\widehat{\kappa}_{p}, c_{p}; \widehat{\vartheta}_{p}, d_{p})} ; X\bigg)\\ && = {}_{p}{\mathbb{B}}_{q}\bigg( {}^{(\kappa_{p}, a_{p}; \vartheta_{p}, b_{p})_{1,p}}_{ (\widehat{\kappa}_{p}, c_{p}; \widehat{\vartheta}_{p}, d_{p})_{1,q}} ; X\bigg)\\ & & = \sum\limits_{k = 0}^{\infty}\dfrac{\bigg\lbrace \prod_{m = 1}^{p}b(\kappa_{m}+a_{m}k; \vartheta_{m}+b_{m}k) \bigg\rbrace}{\bigg\lbrace \prod_{n = 1}^{q}b(\widehat{\kappa}_{n}+c_{n}k; \widehat{\vartheta}_{n}+d_{n}k) \bigg\rbrace}\dfrac{X^{k}}{k!}\\ && = \sum\limits_{k = 0}^{\infty}\frac{\prod_{m = 1}^{p}\bigg( \Gamma (\kappa_{m}+a_{m}k)\Gamma (\vartheta_{m}+b_{m}k) \bigg) \prod_{n = 1}^{q}\Gamma \bigg( (\widehat{\kappa}_{n}+\widehat{\vartheta}_{n})+(c_{n}+d_{n})k \bigg)} {\prod_{m = 1}^{p}\Gamma \bigg( (\vartheta_{m}+\kappa_{m})+(b_{m}+a_{m})k \bigg) \prod_{n = 1}^{q}\bigg( \Gamma(\widehat{\kappa}_{n}+c_{n}k)\Gamma(\widehat{\vartheta}_{n}+d_{n}k) \bigg)}\dfrac{X^{k}}{k!}, \end{eqnarray*}$

where $m, n \in \mathbb{N}, k\in\mathbb{N}\cup \{0\}, \kappa_{m}, \vartheta_{m}, \widehat{\kappa}_{n}, \widehat{\vartheta}_{n}, X\in \mathbb{C}, \: p, q\in \mathbb{N}\cup \{0\},$ and $a_{m}, b_{m}, c_{n}, d_{n}\in \mathbb{R}_{+}.$

We have the following special cases:

$\begin{eqnarray*} {}_{0}\mathbb{B}_{0}& = &e^{X},\\ {}_{1}\mathbb{B}_{0}(X) & = & {}_{1}\mathbb{B}_{0}\bigg( {}^{(\kappa , a; \vartheta, b)}_{\:\:\:-} ; X \bigg)\\ & = &\sum\limits_{k = 0}^{\infty}b(\kappa +ak ; \vartheta+bk)\dfrac{X^{k}}{k!}\\ & = &\sum\limits_{k = 0}^{\infty}\frac{\Gamma(\kappa+ak)\Gamma(\vartheta+bk)}{\Gamma[(b+b)k+(\vartheta+\kappa)]}\dfrac{X^{k}}{k!}\\ & = &{}_{2}\mathbb{W}_{1}\bigg( {}^{(\kappa , b),(\vartheta , b)}_{(\vartheta+\kappa , b+b)} ; X \bigg), \\ {}_{0}\mathbb{B}_{1}(X) & = & {}_{0}\mathbb{B}_{1}\bigg( {}_{(\kappa , a; \vartheta, b)}^{\:\:\:-} ; X \bigg)\\ & = &\sum\limits_{k = 0}^{\infty}\dfrac{1}{b(\kappa +ak ; \vartheta+bk)}\dfrac{X^{k}}{k!}\\ & = &\sum\limits_{k = 0}^{\infty}\dfrac{\Gamma[(b+b)k+(\vartheta+\kappa)]}{\Gamma(\vartheta+bk)\Gamma(\kappa+ak)}\dfrac{X^{k}}{k!}\\ & = &{}_{1}\mathbb{W}_{2}\bigg( {}_{(\kappa , b),(\vartheta , b)}^{(\vartheta+\kappa , b+b)} ; X \bigg), \\ {}_{1}\mathbb{B}_{1}(X) & = & {}_{1}\mathbb{B}_{1}\bigg( {}_{(\kappa , a; \vartheta, b)}^{(\widehat{\kappa}, c ; \widehat{\vartheta},d)} ; X \bigg)\\ & = &\sum\limits_{k = 0}^{\infty}\dfrac{b(\kappa +ak ; \vartheta+bk)}{b(\widehat{\kappa}+ck ; \widehat{\vartheta}+dk)}\dfrac{X^{k}}{k!}\\ & = &\sum\limits_{k = 0}^{\infty}\dfrac{\Gamma(\kappa+ak)\Gamma(\vartheta+bk)\Gamma [ (\widehat{\kappa}+\widehat{\vartheta})+(c+d)k ]}{\Gamma [(b+b)k+(\kappa + \vartheta)] \Gamma(\widehat{\vartheta}+dk) \Gamma (\widehat{\kappa}+ck)}\dfrac{X^{k}}{k!}\\ & = &{}_{3}\mathbb{W}_{3}\bigg( {}^{(\kappa , b), (\vartheta , b), (\widehat{\kappa}+\widehat{\vartheta}, c+d)}_{(\widehat{\kappa},c),(\widehat{\vartheta}, d), (\vartheta+\kappa, b+b)} ; X \bigg), \end{eqnarray*}$

where $k\in\mathbb{N}\cup \{0\}, \kappa_{m}, \vartheta_{m}, \widehat{\kappa}_{n}, \widehat{\vartheta}_{n}, X\in\mathbb{C},$ and $a_{m}, b_{m}, c_{n}, d_{n}\in \mathbb{R}_{+}.$

Now, we define the Wright generalized hypergeometric series (see ^[6]) as follows

$\begin{eqnarray*} {\varphi_{7}^{{\circledS}}}(X): = [{}_{p}\mathbb{W}_{q}]^{n}(X) = \sum\limits_{s = 0}^{n}\dfrac{\bigg\lbrace \prod_{j = 1}^{p}\Gamma(\kappa_{j}+a_{j}s) \bigg\rbrace}{\bigg\lbrace \prod_{k = 1}^{q}\Gamma(\vartheta_{k}+b_{k}s) \bigg\rbrace}\dfrac{X^{s}}{s!}, \end{eqnarray*}$

where $X, \kappa_{j}, \vartheta_{k}\in\mathbb{C}, s, j, k, q, p\in \mathbb{N},$ and $a_{j}, b_{k}\in \mathbb{R}_{+}.$

Let

$\text{diag} [\rho_1,\cdots,\rho_n]_{n\times n} = \begin{bmatrix} \rho_{1} &0 &\ldots & 0 \\ 0 & \rho_{2} &\ddots & \vdots \\ \vdots & \ddots &\ddots& 0 \\ 0&\ldots &0 & \rho_{n} \end{bmatrix}_{n\times n}.$

Note that ${\bf \rho}: = \text{diag} [\rho_1, \cdots, \rho_n]\preceq {\bf \varrho}: = \text{diag} [\varrho_1, \cdots, \varrho_n]$ if $\rho_i\leq \varrho_i$ for each $1\leq i\leq n$ .

We denote $\mathfrak{W}[X]$ as

$\begin{eqnarray*} \text{diag} \bigg[ {\varphi_{1}^{{\circledS}}}(X) , \cdots , {\varphi_{n}^{ {\circledS}}}(X)\bigg]_{n\times n}. \end{eqnarray*}$

A HUR-stability with control functions $\mathfrak{W}[X]$ , is called multi-stability.

3. Investigating multi-stability and super-multi-stability problem associated to (1.1)

We now propose the notion of antiderivations in Banach algebras and introduce the super-multi-stability of antiderivations in algebras Banach, associated to (1.1).

Throughout this section, let $\mathscr{Q}$ be a complex Banach algebra and that $\theta_{1}, \cdots, \theta_{n}\in \mathbb{C}\backslash\{0\}$ with $\vert \theta_{1} \vert, \ldots, \vert \theta_{n} \vert < 1$ .

3.1. Multi stability of the $(\theta_{1}, \cdots, \theta_{n})$ -functional inequality (1.1)

In this subsection, we study the multi stability of the additive $(\theta_{1}, \cdots, \theta_{n})$ -functional inequality (1.1).

Lemma 3.1. Suppose $f_{i}:\mathscr{Q}^{\prime}\times\mathscr{Q}\rightarrow \mathscr{Q} \:(i = 1, \ldots, n\in \mathbb{N})$ are mappings satisfying $f_{i}(\Lambda, 0) = 0$ and (1.1) for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q},$ and $\Lambda\in\mathscr{Q}^{\prime}$ . Then the mappings $f_{i}:\mathscr{Q}^{\prime}\times\mathscr{Q}\rightarrow \mathscr{Q}, \:(i = 1, \ldots, n\in \mathbb{N})$ are additive (the usual definition is at the end of the proof).

Proof. Assume that $f_{i} :\mathscr{Q}^{\prime}\times \mathscr{Q}\rightarrow \mathscr{Q}\:(i = 1, \ldots, n\in \mathbb{N})$ satisfies (1.1).

Replacing $\zeta$ by $-\zeta$ in (1.1), we get

$\begin{eqnarray} & &{\rm diag}\bigg[\:\:\bigg\Vert f_{1} (\Lambda, \varepsilon-\zeta+\varsigma)-f_{1} (\Lambda, \varepsilon+\varsigma)-f_{1} (\Lambda, \varepsilon-\zeta-\varsigma)+f_{1} (\Lambda,\varepsilon-\varsigma)\bigg\Vert , \ldots , \\ &&\:\:\:\:\:\:\:\bigg\Vert f_{n} (\Lambda, \varepsilon-\zeta+\varsigma)-f_{n} (\Lambda, \varepsilon+\varsigma)-f_{n} (\Lambda, \varepsilon-\zeta-\varsigma)+f_{n} (\Lambda,\varepsilon-\varsigma)\bigg\Vert \:\:\bigg]\\ && \preceq \text{diag}\bigg[\:\:\bigg\Vert \theta_{1} \bigg(f_{1} (\Lambda, \varepsilon+\zeta+\varsigma)-f_{1} (\Lambda, \varepsilon+\varsigma)-f_{1} (\Lambda, \varepsilon+\zeta-\varsigma)+f_{1} (\Lambda, \varepsilon-\varsigma)\bigg)\bigg\Vert , \ldots ,\\ &&\:\:\:\:\:\:\:\:\bigg\Vert \theta_{n} \bigg(f_{n} (\Lambda, \varepsilon+\zeta+\varsigma)-f_{n} (\Lambda, \varepsilon+\varsigma)-f_{n} (\Lambda, \varepsilon+\zeta-\varsigma)+f_{n} (\Lambda, \varepsilon-\varsigma)\bigg)\bigg\Vert \:\:\bigg] \end{eqnarray}$

(3.1)

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q},$ and $\Lambda\in\mathscr{Q}^{\prime}$ . According to (1.1) and (3.1) we have

$\begin{align*} &\text{diag}\bigg[ \:\: \bigg\Vert f_{1}(\Lambda , \varepsilon +\zeta+\varsigma)-f_{1}(\Lambda , \varepsilon+\varsigma)-f_{1}(\Lambda , \varepsilon+\zeta-\varsigma)+f_{1}(\Lambda , \varepsilon-\varsigma)\bigg\Vert , \ldots , \nonumber\\ &\notag \:\:\:\:\:\:\:\:\:\:\:\:\:\: \bigg\Vert f_{n}(\Lambda , \varepsilon +\zeta+\varsigma)-f_{n}(\Lambda , \varepsilon+\varsigma)-f_{n}(\Lambda , \varepsilon+\zeta-\varsigma)+f_{n}(\Lambda , \varepsilon-\varsigma)\bigg\Vert \:\:\bigg]\\ &\preceq \text{diag}\bigg[ \:\: \bigg\Vert \theta_{1}^2 \bigg(f_{1}(\Lambda , \varepsilon+\zeta+\varsigma)-f_{1}(\Lambda , \varepsilon+\varsigma)-f_{1}(\Lambda , \varepsilon+\zeta-\varsigma)+f_{1}(\Lambda , \varepsilon-\varsigma)\bigg)\bigg\Vert , \ldots ,\\ &\notag \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \bigg\Vert \theta_{n}^2 \bigg(f_{n}(\Lambda , \varepsilon+\zeta+\varsigma)-f_{n}(\Lambda , \varepsilon+\varsigma)-f_{n}(\Lambda , \varepsilon+\zeta-\varsigma)+f_{n}(\Lambda , \varepsilon-\varsigma)\bigg)\bigg\Vert \:\:\bigg] \end{align*}$

and so

$\begin{equation} f_{i}(\Lambda , \varepsilon +\zeta+\varsigma)-f_{i}(\Lambda , \varepsilon+\varsigma)-f_{i}(\Lambda , \varepsilon+\zeta-\varsigma)+f_{i}(\Lambda , \varepsilon-\varsigma) = 0, \:\:\:\:i = 1,\ldots, n \end{equation}$

(3.2)

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}, \: \Lambda \in \mathscr{Q}^{\prime}$ , since $\vert \theta_{i}\vert < 1\:(i = 1, \ldots, n)$ .

Letting $\varsigma = \varepsilon$ in (3.2),

$\begin{equation*} f_{i}(\Lambda , 2\varepsilon+\zeta)-f_{i}(\Lambda , 2\varepsilon)-f_{i}(\Lambda , \zeta) = 0, \:\:\:\:i = 1,\ldots , n \end{equation*}$

for each $\varepsilon, \zeta \in \mathscr{Q}, \: \Lambda \in \mathscr{Q}^{\prime}$ . Thus $f_{i}\:(i = 1, \ldots, n)$ are additive.

Throughout the paper, let $\varphi_{j_{i}}:(\mathscr{Q})^{3i} \rightarrow [0, \infty)^{i},$ $1\leq i\leq n, \: 1\leq j_{i}\leq n,$ and $n\in\mathbb{N}.$ Notice that $\mathfrak{M}: = \text{diag}[ \varphi_{j_{1}}, \ldots, \varphi_{j_{n}} ]$ is a matrix valued control function such that $\varphi_{j_{1}}(\varphi_{j_{n}})$ represents the element at the $1^{th} (n^{th})$ row and $1^{th} (n^{th})$ column of the matrix $\mathfrak{M}$ and $\varphi_{j_{i}}$ demonstrates the $j_{i}$ th given control function.

Theorem 3.2. Let $(\varphi_{j_{1}}, \ldots, \varphi_{j_{n}}) : (\mathscr{Q}\times \mathscr{Q}\times \mathscr{Q})^{n} \rightarrow [0, \infty)^{n}\:(1\leq j_{1}, \ldots, j_{n}\leq n),$ be functions such that there exists an $(\mathcal{T}_{1}, \ldots, \mathcal{T}_{n}) < \underbrace{(1, \ldots, 1)}_{n}$ with

$\begin{eqnarray} && {\rm diag} \bigg[\:\: \varphi_{j_{1}}(\frac{\varepsilon}{2},\frac{\zeta}{2},\frac{\varsigma}{2}),\ldots , \varphi_{j_{n}}(\frac{\varepsilon}{2},\frac{\zeta}{2},\frac{\varsigma}{2}) \:\:\bigg]\\ &&\preceq {\rm diag} \bigg[\:\: \frac{\mathcal{T}_{1}}{2} \varphi_{j_{1}}(\varepsilon , \zeta , \varsigma),\ldots , \frac{\mathcal{T}_{n}}{2} \varphi_{j_{n}}(\varepsilon , \zeta , \varsigma)\:\:\bigg], \end{eqnarray}$

(3.3)

for all $\varepsilon, \zeta, \varsigma\in \mathscr{Q}$ . Suppose $f_{i}:\mathscr{Q}^{\prime}\times\mathscr{Q}\rightarrow \mathscr{Q}\:(i = 1, \ldots, n)$ are mappings satisfying $f_{i}(\Lambda, 0) = 0$ and

$\begin{align} \label{1-6} &{\rm diag}\bigg[\:\:\bigg\Vert f_{1}(\Lambda,\varepsilon+\zeta+\varsigma)-f_{1}(\Lambda,\varepsilon+\varsigma)-f_{1}(\Lambda,\varepsilon+\zeta-\varsigma)+f_{1}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \\ & \quad + \vert \theta_{1} \vert \underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}(\varepsilon,-\zeta,\varsigma), \ldots , \\ &\:\:\:\:\:\:\:\:\:\:\:\:\bigg\Vert f_{n}(\Lambda,\varepsilon+\zeta+\varsigma)-f_{n}(\Lambda,\varepsilon+\varsigma)-f_{n}(\Lambda,\varepsilon+\zeta-\varsigma)+f_{n}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \\ & \quad + \vert \theta_{n} \vert \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}(\varepsilon,-\zeta,\varsigma)\:\: \bigg]\\ &\preceq {\rm diag}\bigg[\:\:\bigg\Vert \theta_{1} \bigg(f_{1}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{1}(\Lambda,\varepsilon+\varsigma)-f_{1}(\Lambda,\varepsilon-\zeta-\varsigma)+f_{1}(\Lambda,\varepsilon-\varsigma)\bigg)\bigg\Vert \\ & \quad + \underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}(\varepsilon , \zeta, \varsigma), \ldots , \\ &\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\bigg\Vert \theta_{n} \bigg(f_{n}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{n}(\Lambda,\varepsilon+\varsigma)-f_{n}(\Lambda,\varepsilon-\zeta-\varsigma)+f_{n}(\Lambda,\varepsilon-\varsigma)\bigg)\bigg\Vert \\ & \quad + \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}(\varepsilon , \zeta, \varsigma)\:\:\bigg], \end{align}$

(3.4)

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ . Then there exist unique additive mappings $f_{i}^{\prime} :\mathscr{Q}^{\prime}\times \mathscr{Q}\rightarrow \mathscr{Q}$ such that

$\begin{align} & {\rm diag}\bigg[\:\:\bigg\Vert f_{1}(\Lambda ,\varepsilon)-f_{1}^{\prime}(\Lambda,\varepsilon)\bigg\Vert , \ldots , \bigg\Vert f_{n}(\Lambda,\varepsilon)-f_{n}^{\prime}(\Lambda,\varepsilon)\bigg\Vert\:\:\bigg]_{n\times n }\\ &\preceq {\rm diag}\bigg[\:\: \frac{\mathcal{T}_{1}}{2(1-\mathcal{T}_{1})}\underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right),\ldots , \frac{\mathcal{T}_{n}}{2(1-\mathcal{T}_{n})} \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n} \left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right)\:\:\bigg]_{n\times n }, \end{align}$

(3.5)

for each $\varepsilon\in \mathscr{Q}$ , and $\Lambda\in\mathscr{Q}^{\prime}$ .

Proof. Replacing $\zeta$ by $-\zeta$ in (3.4), we get

$\begin{align} &\text{diag}\bigg[\:\:\bigg\Vert f_{1}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{1}(\Lambda,\varepsilon+\varsigma)-f_{1}(\Lambda,\varepsilon-\zeta-\varsigma)+f_{1}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \\ & \quad + \vert \theta_{1} \vert \underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}(\varepsilon,\zeta,\varsigma), \ldots ,\\ & \quad \bigg\Vert f_{n}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{n}(\Lambda,\varepsilon+\varsigma)-f_{n}(\Lambda,\varepsilon-\zeta-\varsigma)+f_{n}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \\ & \quad + \vert \theta_{n} \vert \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}(\varepsilon,\zeta,\varsigma) \:\:\bigg]\\ &\preceq \text{diag}\bigg[\:\:\bigg\Vert \theta_{1} (f_{1}(\Lambda,\varepsilon+\zeta+\varsigma)-f_{1}(\Lambda,\varepsilon+\varsigma)-f_{1}(\Lambda,\varepsilon+\zeta-\varsigma)+f_{1}(\Lambda,\varepsilon-\varsigma))\bigg\Vert\\ & \quad +\underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}(\varepsilon,-\zeta,\varsigma),\ldots , \\ & \quad \bigg\Vert \theta_{n} (f_{n}(\Lambda,\varepsilon+\zeta+\varsigma)-f_{n}(\Lambda,\varepsilon+\varsigma)-f_{n}(\Lambda,\varepsilon+\zeta-\varsigma)+f_{n}(\Lambda,\varepsilon-\varsigma))\bigg\Vert\\ & \quad +\underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}(\varepsilon,-\zeta,\varsigma)\:\:\bigg],\\ \end{align}$

(3.6)

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ . According to (3.4) and (3.6) we have

(3.7)

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ .

Letting $\varepsilon = \varsigma = \frac{\sigma}{2}$ and $\zeta = \sigma$ in (3.7), we get

$\begin{align} & \text{diag}\bigg[\:\:\bigg\Vert f_{1}(\Lambda,2\sigma)-2 f_{1}(\Lambda,\sigma)\bigg\Vert,\ldots , \bigg\Vert f_{n}(\Lambda,2\sigma)-2 f_{n}(\Lambda,\sigma)\bigg\Vert \:\:\bigg]_{n\times n} \\ &\preceq \text{diag}\bigg[\:\:\underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n} \left(\frac{\sigma}{2},\sigma,\frac{\sigma}{2}\right), \ldots , \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}\left(\frac{\sigma}{2},\sigma,\frac{\sigma}{2}\right)\:\:\bigg]_{n\times n}, \end{align}$

(3.8)

for each $\sigma\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ .

Let $\hslash = (\hslash_{1}, \ldots, \hslash_{n})$ and $\hslash^{\prime} = (\hslash^{\prime}_{1}, \ldots, \hslash^{\prime}_{n}).$

Now, consider the set

$\begin{equation*} \nabla: = \{\hslash: (\mathscr{Q}^{\prime}\times\mathscr{Q})^{n}\rightarrow \mathscr{Q}^{n}: \hslash(\Lambda,0) = \overbrace{(0,\ldots , 0)}^{n}\} \end{equation*}$

and define the generalized metric on $\nabla$ by

$\begin{align*} &d(\hbar , \hbar^{\prime}) = \inf \bigg\{(\mu_{1},\ldots , \mu_{n}) \in \Bbb{R}_{+}^{n}: \\ & \quad \text{diag}\bigg[\:\: \Vert \hbar_{1}(\Lambda,\varepsilon)-\hbar_{1}^{\prime}(\Lambda,\varepsilon)\Vert ,\ldots \Vert \hbar_{n}(\Lambda,\varepsilon)-\hbar_{n}^{\prime}(\Lambda,\varepsilon)\Vert \:\:\bigg]\\ & \quad \preceq \text{diag}\bigg[\:\: \mu_{1} \underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right),\ldots , \mu_{n} \underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right)\:\:\bigg], \forall \varepsilon\in \mathscr{Q}, \Lambda\in \mathscr{Q}^{\prime} \bigg\}, \end{align*}$

where $\inf \emptyset = \underbrace{(+\infty, \ldots, +\infty)}_{n}$ .

Now $(\nabla, d)$ is complete (also, see ^[8]).

Let $\mathscr{L}: = (\mathscr{L}_{1}, \ldots, \mathscr{L}_{n}).$ Now, we consider the linear mapping $\mathscr{L}:\nabla \rightarrow \nabla$ s.t.

$\begin{equation*} \mathscr{L}_{i}\hslash_{i}(\Lambda,\varepsilon): = 2\hslash_{i}\left(\Lambda,\frac{\varepsilon}{2}\right),\:\:\:\:i = 1,\ldots , n \end{equation*}$

for each $\varepsilon\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ .

Let $\hslash, \hslash^{\prime}\in \nabla$ be given s.t. $d(\hslash, \hslash^{\prime}) = (\varepsilon_{1}, \ldots, \varepsilon_{n})$ . Then

$\begin{eqnarray*} &\text{diag}\bigg[\:\:\Vert \hslash_{1}(\Lambda,\varepsilon)-\hslash_{1}^{\prime}(\Lambda,\varepsilon)\Vert , \ldots , \Vert \hslash_{n}(\Lambda,\varepsilon)-\hslash_{n}^{\prime}(\Lambda,\varepsilon)\Vert \:\:\bigg]\\ &\preceq \text{diag}\bigg[\:\:\varepsilon_{1} \underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right),\ldots , \varepsilon_{n} \underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right)\:\:\bigg], \end{eqnarray*}$

for each $\varepsilon\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ . Hence

$\begin{align*} &\text{diag}\bigg[\:\: \Vert \mathscr{L}_{1}\hslash_{1}(\Lambda, \varepsilon)-\mathscr{L}_{1}\hslash_{1}^{\prime}(\Lambda,\varepsilon) \Vert , \ldots , \Vert \mathscr{L}_{n}\hslash_{n}(\Lambda, \varepsilon)-\mathscr{L}_{n}\hslash_{n}^{\prime}(\Lambda,\varepsilon) \Vert \:\:\bigg]\\ & = \text{diag}\bigg[\:\: \left\Vert 2\hslash_{1}\left(\Lambda,\frac{\varepsilon}{2}\right)-2\hslash_{1}^{\prime}\left(\Lambda,\frac{\varepsilon}{2}\right)\right\Vert , \ldots , \left\Vert 2\hslash_{n}\left(\Lambda,\frac{\varepsilon}{2}\right)-2\hslash_{n}^{\prime}\left(\Lambda,\frac{\varepsilon}{2}\right)\right\Vert\:\:\bigg]\\ &\preceq \text{diag}\bigg[\:\: 2 \varepsilon_{1} \underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}\left(\frac{\varepsilon}{4},\frac{\varepsilon}{2},\frac{\varepsilon}{4}\right),\ldots , 2 \varepsilon_{n} \underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}\left(\frac{\varepsilon}{4},\frac{\varepsilon}{2},\frac{\varepsilon}{4}\right)\:\:\bigg]\\ &\preceq \text{diag}\bigg[\:\: \mathcal{T}_{1}\varepsilon_{1}\underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right),\ldots , \mathcal{T}_{n}\varepsilon_{n}\underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right)\:\:\bigg], \end{align*}$

for each $\varepsilon\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ . Thus $d(\hslash, \hslash^{\prime}) = \underbrace{(\varepsilon_{1}, \ldots, \varepsilon_{n})}_{n}$ implies that

$d(\mathscr{L}\hslash(\Lambda,\varepsilon),\mathscr{L}\hslash^{\prime}(\Lambda,\varepsilon))\preceq (\mathcal{T}_{1}\varepsilon_{1}, \ldots ,\mathcal{T}_{n}\varepsilon_{n}).$

Hence

$\begin{align*} d(\mathscr{L}\hslash (\varepsilon),\mathscr{L}\hslash^{\prime}(\varepsilon)) \preceq (\mathcal{T}_{1}, \ldots ,\mathcal{T}_{n} ) d(\hslash,\hslash^{\prime}), \end{align*}$

for each $\hslash, \hslash^{\prime}\in \nabla$ . According to (3.8), we get

$\begin{align*} &\text{diag}\bigg[ \:\:\left\Vert f_{1}(\Lambda, \varepsilon)-2f_{1}(\Lambda, \frac{\varepsilon}{2})\right\Vert, \ldots, \left\Vert f_{n}(\Lambda, \varepsilon)-2f_{n}(\Lambda, \frac{\varepsilon}{2})\right\Vert \:\:\bigg]_{n\times n}\\ &\preceq \text{diag}\bigg[ \:\: \underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}\left(\frac{\varepsilon}{4},\frac{\varepsilon}{2},\frac{\varepsilon}{4}\right), \ldots,\underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}\left(\frac{\varepsilon}{4},\frac{\varepsilon}{2},\frac{\varepsilon}{4}\right) \:\:\bigg]_{n\times n}\\ &\preceq \text{diag}\bigg[ \:\: \frac{\mathcal{T}_{1}}{2}\underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right), \ldots, \frac{\mathcal{T}_{n}}{2}\underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right)\:\:\bigg]_{n\times n}, \end{align*}$

for each $\varepsilon\in \mathscr{Q}, \Lambda \in \mathscr{Q}^{\prime}$ , so $d(f, \mathscr{L} f)\preceq (\frac{\mathcal{T}_{1}}{2}, \ldots, \frac{\mathcal{T}_{n}}{2})$ .

According to Theorem 1.2 there exist mappings $f_{i}^{\prime}: \mathscr{Q}\rightarrow \mathscr{Q} (i = 1, \ldots, n)$ satisfying the following:

(1) $f^{\prime}$ is a fixed point of $\mathscr{L}$ , i.e.

$\begin{equation} f^{\prime}(\Lambda, \varepsilon) = 2 f^{\prime} \left(\Lambda, \frac{\varepsilon}{2}\right), \end{equation}$

(3.9)

for each $\varepsilon\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ . The mapping $f^{\prime}$ is a unique fixed point of $\mathscr{L}$ in the set

$\begin{align*} \Bbbk = \{\hslash\in \nabla : d(f,\hslash) < \infty \}. \end{align*}$

This implies that $f^{\prime}$ is a unique mapping satisfying (3.8) s.t. there exist $\mu_{1}, \ldots, \mu_{n} \in (0, \infty)$ satisfying

$\begin{align*} &\text{diag}\bigg[\:\: \Vert f_{1}(\Lambda, \varepsilon)-f_{1}^{\prime}(\Lambda, \varepsilon)\Vert , \ldots , \Vert f_{n}(\Lambda, \varepsilon)-f_{n}^{\prime}(\Lambda, \varepsilon)\Vert \:\:\bigg]\\ & \preceq \text{diag}\bigg[\:\:\mu_{1} \underbrace{\varphi_{j_{1}}}_{0\leq j_{1}\leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right), \ldots , \mu_{n} \underbrace{\varphi_{j_{n}}}_{0\leq j_{n}\leq n}\left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right)\bigg], \end{align*}$

for each $\varepsilon\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ .

(2) Since $\lim_{n\rightarrow \infty}d(\mathscr{L}^{n} f, f^{\prime}) = 0$ ,

$\begin{align} \lim\limits_{n\rightarrow \infty} 2^{n} f_{i}\left(\Lambda,\frac{\varepsilon}{2^{n}}\right) = f_{i}^{\prime}(\Lambda,\varepsilon), \:\: \forall i = 1,\ldots , n \end{align}$

(3.10)

for each $\varepsilon\in \mathscr{Q},$ and $\Lambda\in \mathscr{Q}^{\prime}$ .

(3) $d(f, f^{\prime})\preceq (\frac{1}{1-\mathcal{T}_{1}}, \ldots, \frac{1}{1-\mathcal{T}_{n}})d(f, \mathscr{L}f)$ , which implies

$\begin{align*} &\notag\text{diag}\bigg[\:\:\bigg\Vert f_{1}(\Lambda ,\varepsilon)-f_{1}^{\prime}(\Lambda,\varepsilon)\bigg\Vert , \ldots , \bigg\Vert f_{n}(\Lambda,\varepsilon)-f_{n}^{\prime}(\Lambda,\varepsilon)\bigg\Vert\:\:\bigg]_{n\times n }\\ &\preceq \text{diag}\bigg[\:\: \frac{\mathcal{T}_{1}}{2(1-\mathcal{T}_{1})} \underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n} \left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right),\ldots , \frac{\mathcal{T}_{n}}{2(1-\mathcal{T}_{n})} \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n} \left(\frac{\varepsilon}{2},\varepsilon,\frac{\varepsilon}{2}\right)\:\:\bigg]_{n\times n }, \end{align*}$

for each $\varepsilon\in \mathscr{Q}$ , and $\Lambda\in\mathscr{Q}^{\prime}$ . According to (3.3) and (3.4) we have

$\begin{align*} &\text{diag}\bigg[\:\:\bigg\Vert f_{1}^{\prime}(\Lambda,\varepsilon+\zeta+\varsigma)-f_{1}^{\prime}(\Lambda,\varepsilon+\varsigma)-f_{1}^{\prime}(\Lambda,\varepsilon+\zeta-\varsigma)+f_{1}^{\prime}(\Lambda,\varepsilon-\varsigma)\bigg\Vert\;\: ,\ldots , \\ & \quad \bigg\Vert f_{n}^{\prime}(\Lambda,\varepsilon+\zeta+\varsigma)-f_{n}^{\prime}(\Lambda,\varepsilon+\varsigma)-f_{n}^{\prime}(\Lambda,\varepsilon+\zeta-\varsigma)+f_{n}^{\prime}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \:\:\bigg] \\ & = \text{diag}\bigg[\:\:\lim\limits_{n\rightarrow \infty} 2^{n}\bigg\Vert f_{1}\left(\Lambda,\frac{\varepsilon+\zeta+\varsigma}{2^{n}}\right)- f_{1}\left(\Lambda,\frac{\varepsilon+\varsigma}{2^{n}}\right)\\ & \quad - f_{1}\left(\Lambda,\frac{\varepsilon+\zeta-\varsigma}{2^{n}}\right)+ f_{1} \left(\Lambda,\frac{\varepsilon-\varsigma}{2^{n}}\right) \bigg\Vert\:\:\: ,\ldots , \\ & \quad \lim\limits_{n\rightarrow \infty} 2^{n}\bigg\Vert f_{n}\left(\Lambda,\frac{\varepsilon+\zeta+\varsigma}{2^{n}}\right)- f_{n}\left(\Lambda,\frac{\varepsilon+\varsigma}{2^{n}}\right)\\ & \quad - f_{n}\left(\Lambda,\frac{\varepsilon+\zeta-\varsigma}{2^{n}}\right)+ f_{n} \left(\Lambda,\frac{\varepsilon-\varsigma}{2^{n}}\right) \bigg\Vert \:\:\bigg]\\ &\preceq \text{diag}\bigg[\:\: \lim\limits_{n\rightarrow \infty} 2^{n} \vert \theta_{1} \vert \bigg\Vert f_{1}\left(\Lambda,\frac{\varepsilon-\zeta+\varsigma}{2^{n}}\right)- f_{1}\left(\Lambda,\frac{\varepsilon+\varsigma}{2^{n}}\right)\\ & \quad - f_{1}\left(\Lambda,\frac{\varepsilon-\zeta-\varsigma}{2^{n}}\right)+ f_{1}\left(\Lambda,\frac{\varepsilon-\varsigma}{2^{n}}\right) \bigg\Vert \\ & \quad +\lim\limits_{n\rightarrow \infty} 2^{n}\left(\underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}\left(\frac{\varepsilon}{2^{n}},\frac{\zeta}{2^{n}},\frac{\varsigma}{2^{n}}\right)-\theta_{1} \underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}\left(\frac{\varepsilon}{2^{n}},\frac{-\zeta}{2^{n}},\frac{\varsigma}{2^{n}}\right)\right)\:\: ,\ldots , \\ & \quad \lim\limits_{n\rightarrow \infty} 2^{n} \vert \theta_{n} \vert \bigg\Vert f_{n}\left(\Lambda,\frac{\varepsilon-\zeta+\varsigma}{2^{n}}\right)- f_{n}\left(\Lambda,\frac{\varepsilon+\varsigma}{2^{n}}\right)\\ & \quad - f_{n}\left(\Lambda,\frac{\varepsilon-\zeta-\varsigma}{2^{n}}\right)+ f_{n}\left(\Lambda,\frac{\varepsilon-\varsigma}{2^{n}}\right) \bigg\Vert \\ & \quad +\lim\limits_{n\rightarrow \infty} 2^{n}\left(\underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}\left(\frac{\varepsilon}{2^{n}},\frac{\zeta}{2^{n}},\frac{\varsigma}{2^{n}}\right)-\theta_{n} \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}\left(\frac{\varepsilon}{2^{n}},\frac{-\zeta}{2^{n}},\frac{\varsigma}{2^{n}}\right)\right) \:\:\bigg]\\ &\preceq \text{diag}\bigg[\:\: \bigg\Vert \theta_{n}(f_{1}^{\prime}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{1}^{\prime}(\Lambda,\varepsilon+\varsigma)-f_{1}^{\prime}(\Lambda,\varepsilon-\zeta-\varsigma)+f_{1}^{\prime}(\Lambda,\varepsilon-\varsigma))\bigg\Vert , \ldots ,\\ & \quad \bigg\Vert \theta_{n}(f_{n}^{\prime}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{n}^{\prime}(\Lambda,\varepsilon+\varsigma)-f_{n}^{\prime}(\Lambda,\varepsilon-\zeta-\varsigma)+f_{n}^{\prime}(\Lambda,\varepsilon-\varsigma))\bigg\Vert\:\:\bigg] \end{align*}$

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}$ , and $\Lambda\in\mathscr{Q}^{\prime}$ . According to Lemma 3.1, the mapping $f_{i}^{\prime}\:(i = 1, \ldots, n)$ is additive.

3.2. Super-multi-stability of antiderivations in Banach algebras

Definition 3.3. Assume $\mathscr{Q}$ is a complex Banach algebra. A $\Bbb{C}$ -linear mapping $\mathscr{G} :\mathscr{Q}^{\prime} \times \mathscr{Q} \rightarrow \mathscr{Q}$ is called an antiderivation if it satisfies

$\begin{equation*} \mathscr{G}(\Lambda , \varepsilon)\mathscr{G}(\Lambda , \zeta) = \mathscr{G}(\Lambda ,\mathscr{G}(\Lambda ,\varepsilon)\zeta)+\mathscr{G} (\Lambda , \varepsilon\mathscr{G}(\Lambda ,\zeta)) \end{equation*}$

for each $\varepsilon, \zeta \in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ .

Example 3.4. Suppose $Q_m$ is the collection of all polynomials of degree $m$ with complex coefficients and

$\begin{equation*} Q = \{q_m\in Q_m | q(\Lambda , 0) = 0, m\in\mathbb{N} \}. \end{equation*}$

Define $\mathscr{G}: \mathscr{Q}^{\prime}\times Q \rightarrow Q$ by

$\begin{align*} \mathscr{G}(\Lambda , \sum\limits_{k = 1}^{n}b_{k}\chi^{k}) = \sum\limits_{k = 1}^{n}\frac{b_{k}}{k} \chi^{k} \end{align*}$

and $\mathscr{G}(\Lambda, 0) = 0$ . Then $\mathscr{G}$ is an antiderivation.

Example 3.5. Consider the collection of all continuous functions on $\mathbb{R}$ , represented by $\mathit{C} (\mathbb{R})$ .

Define $\mathscr{G}:\mathscr{Q}^{\prime}\times \mathit{C} (\mathbb{R}) \rightarrow \mathit{C} (\mathbb{R})$ by

$\begin{align*} \mathscr{G}(\Lambda , g(\varepsilon)) = \int_{0}^{\varepsilon}g(\tau)d\tau \end{align*}$

for each $\tau\in \mathbb{R}$ . Then $\mathscr{G}$ is an antiderivation.

Lemma 3.6. ^[9]Suppose $\mathscr{Q}$ is complex Banach algebra and suppose $f:\mathscr{Q}^{\prime}\times\mathscr{Q} \rightarrow \mathscr{Q}$ is an additive mapping s.t. $f(\Lambda, \mathfrak{J}\varepsilon) = \mathfrak{J} f(\Lambda, \varepsilon)$ for each $\mathfrak{J} \in {\mathbb{T}}^1 : = \{ \eta \in {\mathbb{C}} : | \eta| = 1\}$ and each $\varepsilon\in \mathscr{Q}$ . Then $f$ is $\Bbb{C}$ -linear.

Theorem 3.7. Suppose $\varphi_{j_{1}}, \ldots, \varphi_{j_{n}}: \mathscr{Q}^{3} \rightarrow [0, \infty)$ , $(1\leq j_{1}, \ldots, j_{n} \leq n),$ are functions.

(i) If there exist $(\mathcal{T}_{1}, \ldots, \mathcal{T}_{n}) < (1, \ldots, 1)$ satisfying

$\begin{align} &{\rm diag}\bigg[\:\: \underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n} \left(\frac{\varepsilon}{\mathfrak{J} },\frac{\zeta}{\mathfrak{J} },\frac{\varsigma}{\mathfrak{J} }\right) ,\ldots , \underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n} \left(\frac{\varepsilon}{\mathfrak{J} },\frac{\zeta}{\mathfrak{J} },\frac{\varsigma}{\mathfrak{J} }\right)\:\:\bigg] \\ &\preceq{\rm diag}\bigg[\:\:\frac{\mathcal{T}_{1}}{2}\underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}(2\varepsilon,2\zeta,2\varsigma) , \ldots ,\frac{\mathcal{T}_{n}}{2}\underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}(2\varepsilon,2\zeta,2\varsigma)\:\:\bigg], \end{align}$

(3.11)

and if $f_{i}:\mathscr{Q}^{\prime}\times\mathscr{Q}\rightarrow \mathscr{Q}$ , $(i = 1, \ldots, n)$ , are mappings satisfying $f_{i}(\Lambda, 0) = 0$ and

$\begin{align} \label{1-16} &{\rm diag}\bigg[\:\: \bigg\Vert \mathfrak{J} f_{1}(\Lambda,\varepsilon+\zeta+\varsigma)-f_{1}(\Lambda,\mathfrak{J} (\varepsilon+\varsigma))-f_{1}( \Lambda,\mathfrak{J} (\varepsilon+\zeta-\varsigma))+\mathfrak{J} f_{1}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \\ & \quad + \vert \theta_{1} \vert \underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}(\varepsilon,-\zeta,\varsigma) \:\:, \ldots , \\ & \quad \bigg\Vert \mathfrak{J} f_{n}(\Lambda,\varepsilon+\zeta+\varsigma)-f_{n}(\Lambda,\mathfrak{J} (\varepsilon+\varsigma))-f_{n}( \Lambda,\mathfrak{J} (\varepsilon+\zeta-\varsigma))+\mathfrak{J} f_{n}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \\ & \quad + \vert \theta_{n} \vert \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}(\varepsilon,-\zeta,\varsigma)\:\:\bigg]\\ &\preceq {\rm diag}\bigg[\:\: \bigg\Vert \theta_{1} (\mathfrak{J} f_{1}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{1}(\Lambda,\mathfrak{J} (\varepsilon+\varsigma))-f_{1}(\Lambda,\mathfrak{J} (\varepsilon-\zeta-\varsigma))+\mathfrak{J} f_{1}(\Lambda,\varepsilon-\varsigma))\bigg\Vert \\ & \quad +\underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}(\varepsilon,\zeta,\varsigma)\:\:, \ldots , \\ & \quad \bigg\Vert \theta_{n} (\mathfrak{J} f_{n}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{n}(\Lambda,\mathfrak{J} (\varepsilon+\varsigma))-f_{n}(\Lambda,\mathfrak{J} (\varepsilon-\zeta-\varsigma))+\mathfrak{J} f_{n}(\Lambda,\varepsilon-\varsigma))\bigg\Vert \\ & \quad +\underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}(\varepsilon,\zeta,\varsigma)\:\:\bigg], \end{align}$

(3.12)

for each $\mathfrak{J} \in \mathbb{T}^1$ and all $\varepsilon, \zeta, \varsigma\in \mathscr{Q}, \Lambda\in\mathscr{Q}^{\prime}$ , then there exist unique $\Bbb{C}$ -linear mappings $\mathscr{G}_{i}:\mathscr{Q}^{\prime}\times \mathscr{Q}\rightarrow \mathscr{Q}, \:(i = 1, \ldots, n),$ s.t.

$\begin{align} &{\rm diag}\bigg[\:\:\bigg\Vert f_{1}(\Lambda,\varepsilon)-\mathscr{G}_{1}(\Lambda,\varepsilon)\bigg\Vert \: , \ldots , \bigg\Vert f_{n}(\Lambda,\varepsilon)-\mathscr{G}_{n}(\Lambda,\varepsilon)\bigg\Vert \:\:\bigg] \\ &\preceq{\rm diag}\bigg[\:\: \frac{\mathcal{T}_{1}}{2(1-\mathcal{T}_{1})}\underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n} \left(\frac{\varepsilon}{2},\varepsilon ,-\frac{\varepsilon}{2}\right) \: , \ldots , \frac{\mathcal{T}_{n}}{2(1-\mathcal{T}_{n})}\underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n} \left(\frac{\varepsilon}{2},\varepsilon ,-\frac{\varepsilon}{2}\right)\:\:\bigg], \end{align}$

(3.13)

for each $\varepsilon\in \mathscr{Q}, \Lambda\in\mathscr{Q}^{\prime}$ .

(ii) In addition, if $(\mathcal{T}_{1}, \ldots, \mathcal{T}_{n}) < (\frac{1}{2}, \ldots, \frac{1}{2})$ and $f_{i}, (i = 1, \ldots, n),$ are continuous and satisfy $f_{i}(\Lambda, 2\varepsilon) = 2 f_{i}(\Lambda, \varepsilon)$ and

$\begin{align} &{\rm diag}\bigg[\:\: \bigg\Vert f_{1}(\Lambda,\varepsilon)f_{1}(\Lambda,\zeta)-f_{1}(\Lambda,f_{1}(\Lambda,\varepsilon)\zeta)-f_{1}(\Lambda,\varepsilon f_{1}(\Lambda,\zeta))\bigg\Vert \:, \ldots , \\ & \quad \bigg\Vert f_{n}(\Lambda,\varepsilon)f_{n}(\Lambda,\zeta)-f_{n}(\Lambda,f_{n}(\Lambda,\varepsilon)\zeta)-f_{n}(\Lambda,\varepsilon f_{n}(\Lambda,\zeta))\bigg\Vert \:\:\bigg]\\ &\preceq {\rm diag}\bigg[\:\: \underbrace{\varphi_{j_{1}}}_{1\leq j_{1} \leq n}(\varepsilon,\zeta,\varepsilon),\ldots , \underbrace{\varphi_{j_{n}}}_{1\leq j_{n} \leq n}(\varepsilon ,\zeta,\varepsilon)\:\:\bigg], \end{align}$

(3.14)

for each $\varepsilon, \zeta\in \mathscr{Q}$ , then $f_{i}: \mathscr{Q}^{\prime}\times \mathscr{Q}\to \mathscr{Q}$ are antiderivations.

Proof. By a similar method used in Theorem 3.2 the proof of (i) is straightforward. Now, we prove (ii).

(ii) Since $\mathscr{G}_{i} = f_{i}, (i = 1, \ldots, n),$ are continuous and $\mathbb{C}$ -linear, we conclude from (3.11) and (3.14) that

$\begin{align*} &\text{diag}\bigg[\:\: \Vert \mathscr{G}_{1}(\Lambda , \varepsilon)\mathscr{G}_{1}(\Lambda , \zeta)-\mathscr{G}_{1}(\Lambda , \mathscr{G}_{1}(\Lambda , \varepsilon)\zeta)-\mathscr{G}_{1}(\Lambda , \varepsilon\mathscr{G}_{1}(\Lambda , \zeta))\Vert\\ & \quad , \ldots , \Vert \mathscr{G}_{n}(\Lambda , \varepsilon)\mathscr{G}_{n}(\Lambda , \zeta)-\mathscr{G}_{n}(\Lambda , \mathscr{G}_{n}(\Lambda , \varepsilon)\zeta)-\mathscr{G}_{n}(\Lambda , \varepsilon\mathscr{G}_{n}(\Lambda , \zeta))\Vert \:\bigg]\\ & = \text{diag}\bigg[\:\: \lim\limits_{m\rightarrow \infty}4^{m} \bigg\Vert \mathfrak{J}^{m}\bigg(f_{1}\bigg(\Lambda , \frac{\varepsilon}{2^{m}\mathfrak{J}^{m}}\bigg)f_{1}\bigg(\Lambda , \frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)-\mathscr{G}_{1}\bigg(\Lambda , f_{1}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\bigg)\frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)\\ & \quad -\mathscr{G}_{1} \bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}f_{1}\bigg(\Lambda , \frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)\bigg) \bigg)\bigg\Vert \: , \ldots , \\ & \quad \lim\limits_{m\rightarrow \infty}4^{m} \bigg\Vert \mathfrak{J}^{m}\bigg(f_{n}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\bigg)f_{n}\bigg(\Lambda , \frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)-\mathscr{G}_{n}\bigg(\Lambda , f_{n}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\bigg)\frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)\\ & \quad -\mathscr{G}_{n} \bigg(\Lambda , \frac{\varepsilon}{2^{m} \lambda^{m}}f_{n}\bigg(\Lambda , \frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)\bigg) \bigg)\bigg\Vert\: \bigg] \\ & = \text{diag}\bigg[\:\: \lim\limits_{m\rightarrow \infty}4^{m} \bigg\Vert \mathfrak{J}^{m}\bigg(f_{1}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\bigg)f_{1}\bigg(\Lambda , \frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)-f_{1}\bigg(\Lambda , f_{1}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\bigg)\frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)\\ & \quad -f_{1}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}f_{1}\bigg(\Lambda , \frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)\bigg) \bigg)\bigg\Vert , \ldots , \\ & \quad \lim\limits_{m\rightarrow \infty}4^{m} \bigg\Vert \mathfrak{J}^{m}\bigg(f_{n}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\bigg)f_{n}\bigg(\Lambda , \frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)-f_{n}\bigg(\Lambda , f_{1}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\bigg)\frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)\\ & \quad -f_{n}\bigg(\Lambda , \frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}f_{n}\bigg(\Lambda , \frac{\zeta}{2^{m} \mathfrak{J}^{m}}\bigg)\bigg) \bigg)\bigg\Vert \:\:\bigg]\\ &\preceq \text{diag}\bigg[\:\: \lim\limits_{m\rightarrow \infty}2^{2m} \underbrace{\varphi_{j_{1}} }_{1\leq j_{1}\leq n} \left(\frac{\varepsilon}{2^{m} \mathfrak{J}^{m}},\frac{\zeta}{2^{m} \mathfrak{J}^{m}},\frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\right) , \ldots ,\\ & \quad \lim\limits_{m\rightarrow \infty}2^{2m} \underbrace{\varphi_{j_{n}} }_{1\leq j_{n}\leq n} \left(\frac{\varepsilon}{2^{m} \mathfrak{J}^{m}},\frac{\zeta}{2^{m} \mathfrak{J}^{m}},\frac{\varepsilon}{2^{m} \mathfrak{J}^{m}}\right) \:\bigg]\\ &\preceq \text{diag}\bigg[\:\: \lim\limits_{m\rightarrow \infty}(2\mathcal{T}_{1})^{m} \underbrace{\varphi_{j_{1}} }_{1\leq j_{1}\leq n}(\varepsilon,\zeta,\varepsilon) , \ldots , \lim\limits_{m\rightarrow \infty}(2\mathcal{T}_{n})^{m} \underbrace{\varphi_{j_{n}} }_{1\leq j_{n}\leq n}(\varepsilon,\zeta,\varepsilon)\bigg], \end{align*}$

for each $\mathfrak{J}\in \Bbb {T}^1$ and each $\varepsilon, \zeta\in \mathscr{Q}, \Lambda\in\mathscr{Q}^{\prime}$ . Since $\underbrace{(2\mathcal{T}_{1}, \cdots, 2\mathcal{T}_{n})}_{n} < \underbrace{(1, \cdots, 1)}_{n}$ , the $\Bbb{C}$ -linear mappings $\mathscr{G}_{i}, (i = 1, \cdots, n),$ are antiderivations. Thus the mappings $f_{i}: \mathscr{Q}^{\prime}\times \mathscr{Q} \to \mathscr{Q}, (i = 1, \cdots, n),$ are antiderivations.

3.3. Super-multi-stability of continuous antiderivations in Banach algebras

In this subsection, we investigate the super-multi-stability of continuous antiderivations in Banach algebras.

Theorem 3.8. Consider $\underbrace{\varphi_{j_{1}} }_{1 \leq j_{1}\leq n}, \cdots, \underbrace{\varphi_{j_{n}} }_{1 \leq j_{n}\leq n}: \mathscr{Q}^{3} \rightarrow [0, \infty)$ .

(i) If there exist $(\mathcal{T}_{1}, \cdots, \mathcal{T}_{n})\prec \overbrace{(1, \cdots, 1)}^{n}$ satisfying

$\begin{align} &{\rm diag}\bigg[\:\: \underbrace{\varphi_{j_{1}} }_{1 \leq j_{1}\leq n}\left(\frac{\varepsilon}{\mathfrak{J}},\frac{\zeta}{\mathfrak{J}},\frac{\varsigma}{\mathfrak{J}}\right) , \cdots , \underbrace{\varphi_{j_{n}} }_{1 \leq j_{n}\leq n}\left(\frac{\varepsilon}{\mathfrak{J}},\frac{\zeta}{\mathfrak{J}},\frac{\varsigma}{\mathfrak{J}}\right)\:\bigg]\\ &\preceq {\rm diag}\bigg[\:\: \frac{\mathcal{T}_{1}}{2}\underbrace{\varphi_{j_{1}} }_{1 \leq j_{1}\leq n}(2\varepsilon,2\zeta,2\varsigma) , \cdots , \frac{\mathcal{T}_{n}}{2}\underbrace{\varphi_{j_{n}} }_{1 \leq j_{n}\leq n}(2\varepsilon,2\zeta,2\varsigma) \: \bigg] \end{align}$

(3.15)

and if $f_{i}: \mathscr{Q}^{\prime}\times \mathscr{Q}\to \mathscr{Q}, (i = 1, \cdots, n),$ are mappings satisfying $f_{i}(\Lambda, 0) = 0$ and

$\begin{align} \label{1-16c} & {\rm diag}\bigg[\:\:\bigg\Vert \mathfrak{J} f_{1}(\Lambda ,\varepsilon+\zeta+\varsigma)-f_{1}(\Lambda ,\mathfrak{J} (\varepsilon+\varsigma))-f_{1}( \Lambda ,\mathfrak{J}(\varepsilon+\zeta-\varsigma))+\mathfrak{J} f_{1}(\Lambda ,\varepsilon-\varsigma)\bigg\Vert \\ & \quad + \vert \theta_{1} \vert \underbrace{\varphi_{j_{1}} }_{1 \leq j_{1}\leq n}(\varepsilon,-\zeta,\varsigma) \:\:\: , \cdots , \\ & \quad \bigg\Vert \mathfrak{J} f_{n}(\Lambda ,\varepsilon+\zeta+\varsigma)-f_{n}(\Lambda ,\mathfrak{J} (\varepsilon+\varsigma))-f_{n}( \Lambda ,\mathfrak{J}(\varepsilon+\zeta-\varsigma))+\mathfrak{J} f_{n}(\Lambda ,\varepsilon-\varsigma)\bigg\Vert \\ & \quad + \vert \theta_{n} \vert \underbrace{\varphi_{j_{n}} }_{1 \leq j_{n}\leq n}(\varepsilon,-\zeta,\varsigma)\: \bigg]\\ &\preceq {\rm diag}\bigg[\bigg\Vert \theta_{1} (\mathfrak{J}f_{1}(\Lambda ,\varepsilon-\zeta+\varsigma)-f_{1}(\Lambda ,\mathfrak{J} (\varepsilon+\varsigma))-f_{1}(\Lambda ,\mathfrak{J} (\varepsilon-\zeta-\varsigma))+\mathfrak{J} f_{1}(\Lambda ,\varepsilon-\varsigma))\bigg\Vert\\ & \quad + \underbrace{\varphi_{j_{1}} }_{1 \leq j_{1}\leq n}(\varepsilon,\zeta,\varsigma)\:\:\: , \cdots , \\ & \quad \bigg\Vert \theta_{n} (\mathfrak{J}f_{n}(\Lambda ,\varepsilon-\zeta+\varsigma)-f_{n}(\Lambda ,\mathfrak{J} (\varepsilon+\varsigma))-f_{n}(\Lambda ,\mathfrak{J} (\varepsilon-\zeta-\varsigma))+\mathfrak{J} f_{n}(\Lambda ,\varepsilon-\varsigma))\bigg\Vert\\ & \quad + \underbrace{\varphi_{j_{n}} }_{1 \leq j_{n}\leq n}(\varepsilon,\zeta,\varsigma)\: \bigg], \end{align}$

(3.16)

for each $\mathfrak{J} \in \mathbb{C}-\overline{\mathbb{T}}^1$ and each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}, \: \Lambda\in \mathscr{Q}^{\prime}$ , then there are unique $\Bbb{C}$ -linear mappings $\mathscr{G}_{i}: \mathscr{Q}^{\prime}\times \mathscr{Q}\rightarrow \mathscr{Q}, (i = 1, \cdots, n),$ s.t.

$\begin{align} &{\rm diag}\bigg[\:\: \Vert f_{1}(\Lambda ,\varepsilon)-\mathscr{G}_{1}(\Lambda ,\varepsilon)\Vert ,\cdots , \Vert f_{n}(\Lambda ,\varepsilon)-\mathscr{G}_{n}(\Lambda ,\varepsilon)\Vert \bigg]\\ &\preceq{\rm diag}\bigg[\:\: \frac{\mathcal{T}_{1}}{2(1-\mathcal{T}_{1})}\underbrace{\varphi_{j_{1}} }_{1 \leq j_{1}\leq n} \left(\frac{\varepsilon}{2},\varepsilon,-\frac{\varepsilon}{2}\right) ,\cdots , \frac{\mathcal{T}_{n}}{2(1-\mathcal{T}_{n})}\underbrace{\varphi_{j_{n}} }_{1 \leq j_{n}\leq n} \left(\frac{\varepsilon}{2},\varepsilon,-\frac{\varepsilon}{2}\right) \bigg], \end{align}$

(3.17)

for each $\varepsilon\in \mathscr{Q}, \: \Lambda\in \mathscr{Q}^{\prime}$ .

(ii) Furthermore, if $(\mathcal{T}_{1}, \cdots, \mathcal{T}_{n})\prec \overbrace{(\frac{1}{2}, \cdots, \frac{1}{2})}^{n}$ , $\underbrace{\varphi_{j_{1}} }_{1 \leq j_{1}\leq n}, \cdots, \underbrace{\varphi_{j_{n}} }_{1 \leq j_{n}\leq n}$ are continuous functions and also $f_{i}, (i = 1, \cdots, n),$ are continuous and satisfy $f_{i} (\Lambda, 2\varepsilon) = 2 f_{i} (\Lambda, \varepsilon)$ and

$\begin{align*} &{\rm diag}\bigg[\:\:\Vert f_{1}(\Lambda ,\varepsilon)f_{1}(\Lambda ,\zeta)-f_{1}(\Lambda ,f_{1}(\Lambda ,\varepsilon)\zeta)-f_{1}(\Lambda ,\varepsilon f_{1}(\Lambda ,\zeta))\Vert\\ & \quad , \cdots , \Vert f_{n}(\Lambda ,\varepsilon)f_{n}(\Lambda ,\zeta)-f_{n}(\Lambda ,f_{n}(\Lambda ,\varepsilon)\zeta)-f_{n}(\Lambda ,\varepsilon f_{n}(\Lambda ,\zeta))\Vert\;\:\bigg]\\ &\preceq{\rm diag}\bigg[\:\: \underbrace{\varphi_{j_{1}} }_{1 \leq j_{1}\leq n}(\varepsilon,\zeta,\varepsilon) ,\cdots , \underbrace{\varphi_{j_{n}} }_{1 \leq j_{n}\leq n}(\varepsilon,\zeta,\varepsilon)\:\bigg], \end{align*}$

for each $\varepsilon, \zeta\in \mathscr{Q}, \: \Lambda\in \mathscr{Q}^{\prime}$ , then $f_{i}: \mathscr{Q}^{\prime} \times \mathscr{Q} \to \mathscr{Q}$ are continuous antiderivations.

Proof. Using the same reasoning as in the proof of Theorem 3.7, we obtain the desired result.

3.4. Application

Here, let $n = 7.$

Corollary 3.9. Suppose $f_{i}:\mathscr{Q}^{\prime}\times\mathscr{Q}\rightarrow \mathscr{Q}\:(i = 1, \ldots, n)$ are mappings satisfying $f_{i}(\Lambda, 0) = 0$ and

$\begin{align*} &\mathit{\text{diag}}\bigg[\:\:\bigg\Vert f_{1}(\Lambda, \varepsilon+\zeta+\varsigma)-f_{1}(\Lambda, \varepsilon+\varsigma)-f_{1}(\Lambda, \varepsilon+\zeta-\varsigma)+f_{1}(\Lambda, \varepsilon-\varsigma)\bigg\Vert \: , \ldots , \nonumber\\ & \quad \bigg\Vert f_{n}(\Lambda, \varepsilon+\zeta+\varsigma)-f_{n}(\Lambda, \varepsilon+\varsigma)-f_{n}(\Lambda, \varepsilon+\zeta-\varsigma)+f_{n}(\Lambda, \varepsilon-\varsigma)\bigg\Vert \:\:\bigg] \nonumber\\ &\preceq \mathit{\text{diag}}\bigg[\:\: \bigg\Vert \theta_{1} (f_{1}(\Lambda, \varepsilon-\zeta+\varsigma)-f_{1}(\Lambda, \varepsilon+\varsigma)-f_{1}(\Lambda, \varepsilon-\zeta-\varsigma)+f_{1}(\Lambda, \varepsilon-\varsigma))\bigg\Vert\\ & \quad +(1-\vert \theta_{1} \vert){\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}}(\Vert \varepsilon^2\Vert +\Vert \zeta ^2\Vert + \Vert \varsigma^2 \Vert )\: , \ldots , \\ & \quad \bigg\Vert \theta_{n} (f_{n}(\Lambda, \varepsilon-\zeta+\varsigma)-f_{n}(\Lambda, \varepsilon+\varsigma)-f_{n}(\Lambda, \varepsilon-\zeta-\varsigma)+f_{n}(\Lambda, \varepsilon-\varsigma))\bigg\Vert\\ & \quad +(1-\vert \theta_{n} \vert) {\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}(\Vert \varepsilon^2\Vert +\Vert \zeta ^2\Vert + \Vert \varsigma^2 \Vert )\:\:\bigg] \end{align*}$

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}$ , and $\Lambda\in\mathscr{Q}^{\prime}$ . Then there are unique additive mappings $f_{i}^{\prime} :\mathscr{Q}^{\prime}\times \mathscr{Q}\rightarrow \mathscr{Q}$ s.t.

$\begin{align*} &\mathit{\text{diag}}\bigg[ \:\:\Vert f_{1}(\Lambda , \varepsilon)-f_{1}^{\prime}(\Lambda , \varepsilon)\Vert , \ldots , \Vert f_{n}(\Lambda , \varepsilon)-f_{n}^{\prime}(\Lambda , \varepsilon)\Vert \:\:\bigg]_{n\times n}\\ & \preceq \mathit{\text{diag}}\bigg[ \:\: {\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}} (\Vert \varepsilon\Vert ^2 ) , \ldots , {\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}} (\Vert \varepsilon\Vert ^2 )\:\:\bigg], \end{align*}$

for each $\varepsilon\in\mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ .

Proof. The proof follows from Theorem 3.2 by letting

$\begin{align*} &\text{diag}\bigg[ \underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}(\varepsilon, \zeta, \varsigma), \ldots , \underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}(\varepsilon, \zeta, \varsigma)\bigg]\\ &: = \text{diag}\bigg[ {\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}} \bigg(\Vert \varepsilon^2\Vert +\Vert \zeta^2 \Vert + \Vert \varsigma^2 \Vert \bigg),\ldots , {\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}} \bigg(\Vert \varepsilon^2\Vert +\Vert \zeta^2 \Vert + \Vert \varsigma^2 \Vert \bigg)\bigg], \end{align*}$

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}$ . Choosing $(\mathcal{T}_{1}, \ldots, \mathcal{T}_{n}) = (\frac{4}{7}, \ldots, \frac{4}{7}),$ we obtain the desired result.

Corollary 3.10. Suppose $f_{i}:\mathscr{Q}^{\prime}\times\mathscr{Q}\rightarrow \mathscr{Q}, (i = 1, \ldots, n)$ are mappings satisfying $f_{i}(\Lambda, 0) = 0$ and

$\begin{align*} &\mathit{\text{diag}}\bigg[\:\:\bigg\Vert f_{1}(\Lambda, \varepsilon+\zeta+\varsigma)-f_{1}(\Lambda,\varepsilon+\varsigma)-f_{1}(\Lambda,\varepsilon+\zeta-\varsigma)+f_{1}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \:,\ldots , \nonumber\\ & \quad \bigg\Vert f_{n}(\Lambda, \varepsilon+\zeta+\varsigma)-f_{n}(\Lambda,\varepsilon+\varsigma)-f_{n}(\Lambda,\varepsilon+\zeta-\varsigma)+f_{n}(\Lambda,\varepsilon-\varsigma)\bigg\Vert \:\:\bigg]\nonumber\\ &\preceq \mathit{\text{diag}}\bigg[\:\: \bigg\Vert \theta_{1} (f_{1}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{1}(\Lambda,\varepsilon+\varsigma)-f_{1}(\Lambda,\varepsilon-\zeta-\varsigma)+f_{1}(\Lambda,\varepsilon-\varsigma))\bigg\Vert \\ & \quad +(1-\vert \theta_{1} \vert){\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}}(\Vert \varepsilon \zeta\varsigma \Vert) \:\:,\ldots , \\ & \quad \bigg\Vert \theta_{n} (f_{n}(\Lambda,\varepsilon-\zeta+\varsigma)-f_{n}(\Lambda,\varepsilon+\varsigma)-f_{n}(\Lambda,\varepsilon-\zeta-\varsigma)+f_{n}(\Lambda,\varepsilon-\varsigma))\bigg\Vert \\ & \quad +(1-\vert \theta_{n} \vert) {\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}(\Vert \varepsilon \zeta\varsigma \Vert) \:\:\bigg] \end{align*}$

for each $\varepsilon, \zeta, \varsigma \in\mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ . Then there are unique additive mappings $f_{i}^{\prime} : \mathscr{Q}^{\prime}\times\mathscr{Q}\rightarrow \mathscr{Q}, (i = 1, \ldots, n),$ s.t.

$\begin{align*} &\mathit{\text{diag}}\bigg[\:\:\Vert f_{1}(\Lambda , \varepsilon)-f_{1}^{\prime}(\Lambda , \varepsilon)\Vert,\ldots,\Vert f_{n}(\Lambda ,\varepsilon)-f_{n}^{\prime}(\Lambda , \varepsilon)\Vert\:\:\bigg] \\ &\preceq \mathit{\text{diag}}\bigg[\:\: {\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}}(\Vert \varepsilon\Vert ^3 ), \ldots , {\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}(\Vert \varepsilon\Vert ^3 )\:\:\bigg], \end{align*}$

for each $\varepsilon\in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ .

Proof. The proof follows from Theorem 3.2 by letting

$\begin{align*} \text{diag}\bigg[\:\: \underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}(\varepsilon,\zeta, \varsigma), \ldots , \underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}(\varepsilon,\zeta, \varsigma)\:\:\bigg] : = \text{diag}\bigg[\:\: {\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}}(\Vert \varepsilon \zeta \varsigma\Vert),\ldots , {\underbrace{\varphi_{j_{n}}^{ {\circledS}}}_{1\leq j_{n}\leq n}}(\Vert \varepsilon \zeta \varsigma \Vert) \:\:\bigg], \end{align*}$

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ . Choosing $(\mathcal{T}_{1}, \ldots, \mathcal{T}_{n}) = (\frac{8}{9}, \ldots, \frac{8}{9})$ , we obtain the desired result.

Corollary 3.11. Let $f_{i}:\mathscr{Q}^{\prime}\times\mathscr{Q}\rightarrow \mathscr{Q}, (i = 1, \ldots, n)$ be odd mappings satisfying

$\begin{align} &\mathit{\text{diag}}\bigg[\:\: \bigg\Vert f_{1}(\Lambda , \varepsilon +\zeta+\varsigma)-f_{1}(\Lambda , \varepsilon+\varsigma)-f_{1}(\Lambda , \varepsilon+\zeta-\varsigma)+f_{1}(\Lambda , \varepsilon-\varsigma)\bigg\Vert \\ & \quad , \ldots , \bigg\Vert f_{n}(\Lambda , \varepsilon +\zeta+\varsigma)-f_{n}(\Lambda , \varepsilon+\varsigma)-f_{n}(\Lambda , \varepsilon+\zeta-\varsigma)+f_{n}(\Lambda , \varepsilon-\varsigma)\bigg\Vert \:\:\bigg] \\ &\preceq \mathit{\text{diag}}\bigg[\:\: \bigg\Vert \theta_{1} [f_{1}(\Lambda , \varepsilon-\zeta+\varsigma)-f_{1}(\Lambda , \varepsilon+\varsigma)-f_{1}(\Lambda , \varepsilon-\zeta-\varsigma)+f_{1}(\Lambda , \varepsilon-\varsigma)]\bigg\Vert \\ & \quad +(1-\vert \theta_{1} \vert) {\underbrace{\varphi_{j_{1}}^{ {\circledS}}}_{1\leq j_{1}\leq n}}(\Vert \varepsilon\zeta\varsigma\Vert )\\ & \quad , \ldots , \:\: \bigg\Vert \theta_{n} [f_{n}(\Lambda , \varepsilon-\zeta+\varsigma)-f_{n}(\Lambda , \varepsilon+\varsigma)-f_{n}(\Lambda , \varepsilon-\zeta-\varsigma)+f_{n}(\Lambda , \varepsilon-\varsigma)]\bigg\Vert \\ & \quad +(1-\vert \theta_{n} \vert) {\underbrace{\varphi_{j_{n}}^{ {\circledS}}}_{1\leq j_{n}\leq n}}(\Vert \varepsilon\zeta\varsigma\Vert ) \:\: \bigg], \end{align}$

(3.18)

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ . Then $f_{i}(i = 1, \ldots, n)$ are additive.

Proof. Putting $\varepsilon = 0$ in (3.18), we get

$\begin{align} &\text{diag}\bigg[\:\: \bigg\Vert f_{1}(\Lambda , \zeta+\varsigma)-f_{1}(\Lambda , \varsigma)-f_{1}(\Lambda, \zeta-\varsigma)+f_{1}(\Lambda , -\varsigma)\bigg\Vert \:\:, \ldots , \\ & \quad \bigg\Vert f_{n}(\Lambda , \zeta+\varsigma)-f_{n}(\Lambda , \varsigma)-f_{n}(\Lambda , \zeta-\varsigma)+f_{n}(\Lambda , -\varsigma)\bigg\Vert \:\:\bigg]\\ &\preceq \text{diag}\bigg[\:\: \bigg\Vert \theta_{1} (f_{1}(\Lambda , -\zeta+\varsigma)-f_{1}(\Lambda , \varsigma)-f_{1}(\Lambda , -\zeta-\varsigma)+f_{1}(\Lambda , -\varsigma))\bigg\Vert \:\: , \ldots ,\\ & \quad \bigg\Vert \theta_{n} (f_{n}(\Lambda , -\zeta+\varsigma)-f_{n}(\Lambda , \varsigma)-f_{n}(\Lambda , -\zeta-\varsigma)+f_{n}(\Lambda , -\varsigma))\bigg\Vert \:\:\bigg], \end{align}$

(3.19)

for each $\zeta, \varsigma\in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ . Replacing $\zeta$ by $-\zeta$ in (3.19), we have

$\begin{align} &\text{diag}\bigg[\:\:\bigg\Vert f_{1}(\Lambda , -\zeta+\varsigma)-f_{1}(\Lambda , \varsigma)-f_{1}(\Lambda , -\zeta-\varsigma)+f_{1}(\Lambda , -\varsigma)\bigg\Vert \:\:, \ldots , \\ & \quad \bigg\Vert f_{n}(\Lambda , -\zeta+\varsigma)-f_{n}(\Lambda , \varsigma)-f_{n}(\Lambda , -\zeta-\varsigma)+f_{n}(\Lambda , -\varsigma)\bigg\Vert \:\:\bigg] \\ &\preceq \text{diag}\bigg[\:\: \bigg\Vert \theta_{1} [f_{1}(\Lambda , \zeta+\varsigma)-f_{1}(\Lambda , \varsigma)-f_{1}(\Lambda , \zeta-\varsigma)+f_{1}(\Lambda , -\varsigma)]\bigg\Vert \:\: , \ldots , \\ & \quad \bigg\Vert \theta_{n} [f_{n}(\Lambda , \zeta+\varsigma)-f_{n}(\Lambda , \varsigma)-f_{n}(\Lambda , \zeta-\varsigma)+f_{n}(\Lambda , -\varsigma)]\bigg\Vert \:\: \bigg], \end{align}$

(3.20)

for each $\zeta, \varsigma\in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ . From (3.19) and (3.20), it follows that

$\begin{equation*} f_{i}(\Lambda , \zeta+\varsigma)-f_{i}(\Lambda , \varsigma)-f_{i}(\Lambda , \zeta-\varsigma)+f_{i}(\Lambda , -\varsigma) = 0,\:\:\:\:\:i = 1,\ldots , n \end{equation*}$

for each $\zeta, \varsigma\in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ . Since $f_{i}, (i = 1, \ldots, n)$ , are odd mappings,

$\begin{equation*} f_{i}(\varsigma+\zeta)+f_{i}(\varsigma-\zeta)-2f_{i}(\varsigma) = 0,\:\:\:\:\:i = 1,\ldots , n \end{equation*}$

for each $\zeta, \varsigma\in \mathscr{Q}$ and $\Lambda\in\mathscr{Q}^{\prime}$ . Thus the mappings $f_{i}, (i = 1, \ldots, n)$ , are additive.

Corollary 3.12. Suppose $f_{i}: \mathscr{Q}^{\prime}\times \mathscr{Q} \to \mathscr{Q}, (i = 1, \cdots, n),$ are mappings satisfying $f_{i}(\Lambda, 0) = 0$ and

$\begin{align*} &\mathit{\text{diag}}\bigg[\:\:\bigg\Vert f_{1}(\Lambda, \varepsilon+\zeta+\varsigma)- f_{1}(\Lambda, \varepsilon+\varsigma)- f_{1}(\Lambda, \varepsilon+\zeta-\varsigma)+ f_{1}(\Lambda, \varepsilon-\varsigma)\bigg\Vert , \cdots , \\ & \quad \bigg\Vert f_{n}(\Lambda, \varepsilon+\zeta+\varsigma)- f_{n}(\Lambda, \varepsilon+\varsigma)- f_{n}(\Lambda, \varepsilon+\zeta-\varsigma)+ f_{n}(\Lambda, \varepsilon-\varsigma)\bigg\Vert \:\:\bigg]\nonumber\\ &\preceq \mathit{\text{diag}}\bigg[\:\: \bigg\Vert \theta_{1} ( f_{1}(\Lambda, \varepsilon-\zeta+\varsigma)- f_{1}(\Lambda, \varepsilon+\varsigma)- f_{1}(\Lambda, \varepsilon-\zeta-\varsigma)+ f_{1}(\Lambda, \varepsilon-\varsigma))\bigg\Vert \\ & \quad +(1-\vert \theta_{1} \vert) {\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}}\bigg(\Vert \varepsilon^{4}+\zeta^{4}+\varsigma^{4} \Vert\bigg)\:\: , \cdots , \\ & \quad \bigg\Vert \theta_{n} ( f_{n}(\Lambda, \varepsilon-\zeta+\varsigma)- f_{n}(\Lambda, \varepsilon+\varsigma)- f_{n}(\Lambda, \varepsilon-\zeta-\varsigma)+ f_{n}(\Lambda, \varepsilon-\varsigma))\bigg\Vert \\ & \quad +(1-\vert \theta_{n} \vert){\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}\bigg( \Vert \varepsilon^{4}+\zeta^{4}+\varsigma^{4} \Vert\bigg) \:\:\bigg] \end{align*}$

and

$\begin{align*} &\mathit{\text{diag}}\bigg[\:\:\bigg\Vert f_{1}(\Lambda, \varepsilon)f_{1}(\Lambda, \zeta)-f_{1}(\Lambda, f_{1}(\Lambda, \varepsilon)\zeta)-f_{1}(\Lambda, \varepsilon f_{1}(\Lambda, \zeta))\bigg\Vert \: , \cdots , \\ & \quad \bigg\Vert f_{n}(\Lambda, \varepsilon)f_{n}(\Lambda, \zeta)-f_{n}(\Lambda, f_{n}(\Lambda, \varepsilon)\zeta)-f_{n}(\Lambda, \varepsilon f_{n}(\Lambda, \zeta))\bigg\Vert \: \bigg] \\ & \preceq \mathit{\text{diag}}\bigg[\:\: {\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}}\bigg(\Vert 2\varepsilon^{4}+\zeta^{4}\Vert\bigg) , \cdots , {\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}\bigg(\Vert 2\varepsilon^{4}+\zeta^{4}\Vert \bigg)\bigg]_{n\times n}, \end{align*}$

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}, \: \Lambda\in \mathscr{Q}^{\prime}$ . If $f_{i}(\Lambda, 2\varepsilon) = 2 f_{i}(\Lambda, \varepsilon)$ foreach $\varepsilon, \zeta, \varsigma\in \mathscr{Q}, \: \Lambda\in \mathscr{Q}^{\prime}$ , and $f_{i}, (i = 1, \cdots, n),$ are continuous, then the mappings $f_{i}: \mathscr{Q}^{\prime}\times \mathscr{Q}\to \mathscr{Q}, (i = 1, \cdots, n),$ are antiderivations.

Proof. The proof follows from Theorem 3.7 by letting

$\begin{eqnarray*} &&\text{diag}\bigg[\:\: \underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}(\varepsilon , \zeta , \varsigma) , \cdots , \underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}(\varepsilon , \zeta , \varsigma) \bigg]\\ &&: = \text{diag}\bigg[\:\:{\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}} \bigg(\Vert \varepsilon^{4}+\zeta^{4}+\varsigma^{4} \Vert\bigg) \: , \cdots , \:{\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}\bigg( \Vert \varepsilon^{4}+\zeta^{4}+\varsigma^{4} \Vert \bigg) \:\bigg]_{n\times n} \end{eqnarray*}$

for each $\varepsilon, \zeta, \varsigma\in \mathscr{Q}$ . Choosing $(\mathcal{T}_{1}, \cdots, \mathcal{T}_{n}) = \overbrace{(\frac{8}{17}, \cdots, \frac{8}{17})}^{n}$ , we obtain the desired result.

4. Conclusions

In this study, we investigated the concept of antiderivations in Banach algebras and study multi-super-stability of antiderivations in Banach algebras, associated with functional inequalities.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no RP-21-09-08.

Conflict of interest

The authors declare that they have no competing interests.

Acknowledgments

The author thanks Johnny Kristiansen and Pia Frostad for their assistance in the field and laboratory. The study was supported by NMBU grant 1750051444.

Conflict of interest

The author declares no conflict of interests.

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AIMS Microbiology

Hypolimnetic assimilation of ammonium by the nuisance alga Gonyostomum semen

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Abstract

1. Introduction

2. Proposing new control functions and the concept of multi stability

3. Investigating multi-stability and super-multi-stability problem associated to (1.1)

3.1. Multi stability of the $(\theta_{1}, \cdots, \theta_{n})$ -functional inequality (1.1)

3.2. Super-multi-stability of antiderivations in Banach algebras

3.3. Super-multi-stability of continuous antiderivations in Banach algebras

3.4. Application

4. Conclusions

Acknowledgments

Conflict of interest

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AIMS Microbiology

Hypolimnetic assimilation of ammonium by the nuisance alga Gonyostomum semen

Related Papers:

Abstract

1. Introduction

2. Proposing new control functions and the concept of multi stability

3. Investigating multi-stability and super-multi-stability problem associated to (1.1)

3.1. Multi stability of the (θ1,⋯,θn) (\theta_{1}, \cdots, \theta_{n}) -functional inequality (1.1)

3.2. Super-multi-stability of antiderivations in Banach algebras

3.3. Super-multi-stability of continuous antiderivations in Banach algebras

3.4. Application

4. Conclusions

Acknowledgments

Conflict of interest

References

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3.1. Multi stability of the $(\theta_{1}, \cdots, \theta_{n})$ -functional inequality (1.1)