Multi-super-stability of antiderivations in Banach algebras

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

and also

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

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

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

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

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:

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

Euler's functional equation is given by

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

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

The Pochhammer symbol is

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

Note that

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

The Gauss hypergeometric series ^[7] is given by

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

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

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

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

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

where

and

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]:

and

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

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

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

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

where $\Xi+1\geq 0.$

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

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:

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

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

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

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

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

and so

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),

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

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

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

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

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

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

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

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

and define the generalized metric on $\nabla$ by

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.

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

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

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

Hence

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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.

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

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

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.

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

Proof. The proof follows from Theorem 3.2 by letting

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

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.

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

Proof. The proof follows from Theorem 3.2 by letting

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

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

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

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

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

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

and

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

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.