Refined stability of the additive, quartic and sextic functional equations with counter-examples

1.   Introduction and basic facts

In many different settings, functional equations are essential to the investigation of stability problems. The first in challenging the stability of group homomorphisms was Ulam ^[1]. His work laid the groundwork for subsequent research on stability phenomena. If an equation allows only one unique solution, we refer to that equation as being stable. Ulam ^[1] formulated the following Cauchy functional equation:

In the context of a Banach space, Hyers ^[2] addressed Cauchy's functional equation in order to resolve this problems. Aoki ^[3] improved the work of Hyers by taking an unbounded Cauchy difference. Rassias ^[4] discussed additive mappings in his study, and Găvruţa ^[5] has already given identical results. For more details about the stability results, see ^{[2,6,7,8,9,10,11,12,13]}.

In 1950, Nakano ^[14] investigated the idea of modular linear spaces. Numerous writers have now extensively verified these hypotheses, e.g., Luxemburg ^[15], Amemiya ^[16], Musielak ^[17], Koshi ^[18], Mazur ^[19], Turpin ^[20] and Orlicz ^[21]. Both Orlicz spaces ^[22] and the concept of interpolation ^[17,22] have several applications in the setting of modular spaces.

Several researchers examined stability in modular spaces via a fixed point approach of quasi-contractions without utilizing the $\Delta _{2}$-condition, as suggested by Khamsi ^[23]. In recent years, Sadeghi ^[24] produced results on stability of some functional equations combining the $\Delta _{2}$-condition with the Fatou property.

First, we review some terminology, notations, and common characteristics of the theory of given spaces.

Definition 1.1. ^[24] Let $Q$ be a linear space over $\Bbbk$ $(\mathbb{R}$ or $\mathbb{C}).$ A function $\varrho :Q\rightarrow \lbrack 0, \infty)$ is said to be modular if the hypotheses below hold for all $\varpi, \rho \in Q$:

$(m_{1})$ $\varrho (\varpi) = 0\Leftrightarrow \varpi = 0;$

$(m_{2})$ $\varrho (a\varpi) = \varrho (\varpi)$ for any scalar $a$ with $\left\vert a\right\vert = 1;$

$(m_{3})$ $\varrho (a_{1}\varpi +a_{2}\rho)\leq \varrho (\varpi)+\varrho (\rho)$ for any scalar $a_{1}, a_{2}\geq 0$ with $a_{1}+a_{2} = 1.$

Also, $\varrho$ is said to be convex modular, if the hypothesis $(m_{3})$ is replaced by

$(m_{3}^{\prime })$ $\varrho (a_{1}\varpi +a_{2}\rho)\leq a_{1}\varrho (\varpi)+a_{2}\varrho (\rho)$ for any scalar $a_{1}, a_{2}\geq 0$ with $a_{1}+a_{2} = 1.$

Additionally, the vector space induced by a modular $\varrho,$

is a modular space (MS, for short). Denote by $\mathbb{N}$ the set of positive integers.

Definition 1.2. ^[24] Let $\{\varpi _{\mu }\}$ be a sequence in an MS $Q_{\varrho }.$

(i) If $\varrho (\varpi _{j }-\varpi)\rightarrow 0$ as $j \rightarrow \infty,$ then $\{\varpi _{j}\}$ is called $\varrho$-convergent to a point $\varpi$ and we write $\varpi _{j}\rightarrow \varpi$ as $j \rightarrow \infty.$

(ii) If $\varrho (\varpi _{j }-\varpi _{\xi }) < \epsilon$ for any $\epsilon > 0$ and for sufficiently large $j, \xi \in \mathbb{N},$ then $\{\varpi _{j}\}$ is called $\varrho$-Cauchy.

(iii) $K\subseteq Q_{\varrho }$ is called $\rho$-complete if any $\varrho$-Cauchy sequence is $\varrho$-convergent.

(iv) A modular $\varrho$ is said to satisfy the $\Delta _{2}$-condition if $\varrho (2\varpi _{j})\rightarrow 0$ as $j\rightarrow \infty,$ whenever $\varrho (\varpi _{j})\rightarrow 0$ as $j\rightarrow \infty.$

If $\varrho (\varpi)\leq \lim \inf_{\mu \rightarrow \infty }\varrho (\varpi _{\mu}),$ the modular $\varrho$ possesses the Fatou property, whereas the sequence $\{\varpi _{\mu }\}$ is $\varrho$-convergent to $\varpi$ in the MS $Q_{\varrho }$ and vice versa.

Proposition 1.1. ^[25] In MSs,

(1) if $\varpi _{\alpha }\rightarrow \varpi$ and $\lambda$ is a constant vector, then $\varpi _{\alpha }+\lambda \rightarrow \varpi +\lambda;$

(2) if $\varpi _{\alpha }\rightarrow \varpi$ and $\rho _{\alpha }\rightarrow \rho,$ then $a_{1}\varpi _{\alpha }+a_{2}\rho _{\alpha }\rightarrow a_{1}\varpi +a_{2}\rho,$ where $a_{1}, a_{2}\geq 0$ and $a_{1}+a_{2}\leq 1.$

Remark 1.1. Suppose that $\rho$ is convex and justifies the $\Delta _{2}$-condition with $\Delta _{2}$-constant $r > 0$. If $r < 2$, then $\varrho (\varpi)\leq r\varrho (\frac{\varpi }{2})\leq \frac{r}{2}\varrho (\varpi),$ which suggests $\varrho = 0.$ Therefore, if $\rho$ is a convex modular, we ought to obtain the $\Delta _{2}$-constant $r\geq 2.$

It is clear that if $\mu$ is chosen from the analogous scalar field with $\left\vert \mu \right\vert > 1$ in MSs, then the convergence of a sequence $\{\varpi _{\alpha }\}$ to $\varpi$ does not imply that $\{\mu \varpi _{\alpha }\}$ converges to $\mu \varpi.$ This is due to the fact that in MSs, the multiples of the convergent sequence $\{\varpi _{\alpha }\}$ are convergent naturally.

In 1960, the idea of linear $2$-normed spaces was created by Gahler ^[26] as follows:

Definition 1.3. Assume that $\Lambda$ over $\mathbb{R}$ is a linear space with $\dim \Lambda > 1$ and a function $\left\Vert., .\right\Vert :\Lambda \times \Lambda \rightarrow \mathbb{R}$ is such that for all $\varpi, \rho, \ell \in \Lambda$ and $\vartheta \in \mathbb{R},$

(i) $\left\Vert \varpi, \rho \right\Vert = 0$ iff $\varpi$ and $\rho$ are linearly dependent;

(ii) $\left\Vert \varpi, \rho \right\Vert = \left\Vert \rho, \varpi \right\Vert;$

(iii) $\left\Vert \vartheta \varpi, \rho \right\Vert = \left\vert \vartheta \right\vert \left\Vert \varpi, \rho \right\Vert;$

(iv) $\left\Vert \varpi, \ell +\rho \right\Vert \leq \left\Vert \varpi, \ell \right\Vert +\left\Vert \varpi, \rho \right\Vert.$

Then the function $\left\Vert., .\right\Vert$ is called a $2$-norm on $\Lambda$, and $(\Lambda, \left\Vert., .\right\Vert)$ is called a linear $2$-normed space ($2$-NS, for short).

For example of a $2$-NS, consider $\mathbb{R} ^{2}$ endowed with a $2$-norm defined by $\left\vert \varpi -\rho \right\vert =$the area of the triangle with vertices $0,$ $\varpi$ and $\rho.$

It should be noted that, the assertion $(iv)$ implies that

Hence, the mapping $\varpi \rightarrow \left\Vert \varpi, \rho \right\Vert$ is continuous from $\Lambda$ onto $\mathbb{R},$ for any fixed $\rho \in \mathbb{R}.$

Definition 1.4. Let $\Lambda$ be a linear $2$-NS and $\{\varpi _{j}\}_{j\geq 1}$ be a sequence in $\Lambda.$

(1) A sequence $\{\varpi _{j}\}_{j\geq 1}$ is called convergent if there exists an element $\varpi \in \Lambda$ such that

If $\{\varpi _{j}\}_{j\geq 1}$ converges to $\varpi,$ then we can write $\varpi _{j}\rightarrow \varpi$ as $j\rightarrow \infty$ or $\lim_{j\rightarrow \infty }\varpi _{j} = \varpi$ and we say that $\varpi$ is a limit point of $\{\varpi _{j}\}_{j\geq 1}.$

(2) Assume that $\ell, \rho \in \Lambda$ such that $\ell$ and $\rho$ are linearly independent. Then $\{\varpi _{j}\}_{j\geq 1}$ is called a Cauchy sequence in $\Lambda$, if

and

Definition 1.5. A linear 2-NS in which every Cauchy sequence is a convergent sequence is called a 2-Banach space (2-BS, for short).

Lemma 1.1. ^[27] Assume that $(\Lambda, \left\Vert., .\right\Vert)$ is a $2$-NS. If $\varpi \in \Lambda$ and $\left\Vert \varpi, \rho \right\Vert = 0$ for each $\rho \in \Lambda,$ then $\rho = 0.$

Lemma 1.2. ^[27] Let $\{\varpi _{j}\}_{j\geq 1}$ be a convergent sequence in a linear $2$-NS $\Lambda,$ then,

Sadeghi ^[24] has confirmed the stability findings of functional equations utilizing the Fatou property and the $\Delta _{2}$-condition in modular spaces. Our paper is aimed to discuss the refined stability of additive, quartic and sextic functional equations

and

respectively, in MSs with and without the $\Delta _{2}$-condition and by the direct technique. Additionally, the Ulam stability in $2$-BSs is examined. Finally, we show that the stability of these equations does not hold in a particular scenario using appropriate counter-examples.

2.   Stability analysis

Here, we apply the direct technique to examine the stability theorems of the additive, quartic, and sextic functional equation. These results are considered as an improvement of forms due to Wongkum ^[28] and Sadeghi ^[24]. We assume here $\Lambda$ is a linear space and $Q_{\varrho }$ is a complete convex MS.

2.1.   Additive functional equation and stability study

Kim ^[29] in 2013 studied the stability of the additive functional equation in fuzzy BSs. Inspired by the technique of Kim ^[29], we aim to study the stability of the additive functional equation:

for any $m > 0$ in modular spaces by ignoring the conditions of $\Delta _{2}.$ For the convenience of notation, define the mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ as

where $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda,$ and $m$ is a fixed nonzero integer.

Theorem 2.1. Assume that there is a function $\Xi :\Lambda ^{4}\rightarrow \lbrack 0, \infty)$ defined by

such that a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ satisfies $\Omega (0) = 0$ and for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda,$

Then there exists a unique additive mapping (AM) $W:\Lambda \rightarrow Q_{\varrho }$ fulfilling

Proof. Putting $s_{1} = s_{2} = s_{3} = s_{4}$ in (2.3), and setting $\Xi \left(s_{1}, s_{1}, s_{1}, s_{1}\right) = U(s_{1}),$ we have

Hence,

Based on a mathematical induction, one can deduce that

for all $s_{1}\in \Lambda$ and all natural numbers $\mu.$ Clearly, (2.6) follows immediately form (2.7) if we take $\mu = 1.$ Assume that the inequality (2.7) is true for $\mu \in \mathbb{N},$ then we get

It follows that the inequality (2.7) is true for every $\mu \in \mathbb{N}.$ Suppose that $\theta$ and $\eta$ are natural numbers with $\theta < \eta.$ Using (2.7), we can write

Inequalities (2.2) and (2.8) illustrate that $\left\{ \frac{ \Omega \left(4^{\eta }s_{1}\right) }{4^{\eta }}\right\}$ is $\varrho$-Cauchy sequence in $Q_{\varrho }.$ Since $Q_{\varrho }$ is $\varrho$-complete, one can say $\left\{ \frac{\Omega \left(4^{\eta }s_{1}\right) }{4^{\eta } }\right\}$ is $\varrho$-convergent. Now, describe the mapping $W:\Lambda \rightarrow Q_{\varrho }$ as

Hence,

for all $s_{1}\in \Lambda.$ Applying (2.9) in (2.10) after taking the limit as $\eta \rightarrow \infty,$ we find that the right-hand side of (2.10) tends to $0$. Thus, one gets

Also, for all $\eta \in \mathbb{N},$ by (2.11), we observe that

Since $\sum\limits_{\mu = 1}^{\eta }\frac{1}{4^{\mu }}+\frac{1}{4} < 1,$ by (2.5) and (2.12), one can write

Passing to the limit as $\eta \rightarrow \infty$ in (2.13), we have

Therefore, the inequality (2.4) is true. Now, we shall prove that $W$ is an AM. It is easy to observe that

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda.$ When $j\rightarrow \infty$ in (2.14), we get $\varrho \left(\Delta W\left(s_{1}, s_{2}, s_{3}, s_{4}\right) \right) \rightarrow 0.$ Hence,

This implies that $W$ is an additive mapping. For the uniqueness, assume that $W_{1}$ and $W_{2}$ are two AMs that satisfy (2.4). Then,

which yields that $W_{1} = W_{2}.$ This finishes the proof. □

The corollaries below follow immediately from Theorem 2.1:

Corollary 2.1. If there exists a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that $\Omega (0) = 0$ and

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda,$ then there exists a unique AM $W:\Lambda \rightarrow Q_{\varrho }$ satisfying

Corollary 2.2. If there exists a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that $\Omega (0) = 0$ and

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda,$ $\xi > 0$ and $p\in (0, 1),$ then there exists a unique AM $W:\Lambda \rightarrow Q_{\varrho }$ fulfilling

In the context of MSs, we present another stability result as in Theorem 2.1 with condition $\Delta _{2}$ as follows:

Theorem 2.2. Let $Q$ be a linear space and $Q_{\varrho }$ fulfill the $\Delta _{2}$-condition with the mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that

and

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda.$ Then there exists a unique AM $W:\Lambda \rightarrow Q_{\varrho },$ described as

and

for all $s_{1}\in \Lambda.$

Proof. Since $\varrho$ verifies the $\Delta _{2}$-condition with $\alpha,$ Eq (2.3) implies that

So, the proof of Theorem 2.1 directly leads to the conclusion. □

2.2.   Quartic functional equation and stability study

In this part, without using the Fatou property, the refined and Ulam stability of the following quartic functional equation are investigated:

in modular spaces $Q_{\varrho }.$ For ease of notations, we can define a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ as

for all $s_{1}, s_{2}\in \Lambda.$

Theorem 2.3. Let $Q$ be a linear space and $Q_{\varrho }$ fulfill the $\Delta _{2}$-condition with a mapping $\Xi :\Lambda \times \Lambda \rightarrow \lbrack 0, \infty)$. Suppose also there exists a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that

for all $s_{1}, s_{2}\in \Lambda.$ Then there exists a unique quartic mapping (QM) $W:\Lambda \rightarrow Q_{\varrho }$ described as

and

for all $s_{1}\in \Lambda.$

Proof. Consider $\Omega (0) = 0$ in view of $\Xi \left(0, 0\right) = 0$ along the convergence of

Letting $s_{2} = 0$ in (2.16), we have

Since $\sum\limits_{j = 1}^{\infty }\frac{1}{2^{j}} < 1,$ based on $\Delta _{2}$-condition of $\varrho,$ the subsequent functional inequality can be written as

Now, in (2.18), replacing $s_{1}$ with $\frac{s_{1}}{3^{\mu }},$ we conclude that the series in (2.16) converges, and

for all $s_{1}\in \Lambda.$ Since $\frac{3^{\theta }}{u^{\theta +3}}\leq 1,$ the right-hand side of the above inequality tends to $0.$ This proves that $\left\{ 3^{4\mu }\Omega \left(\frac{s_{1}}{3^{\mu }}\right) \right\}$ is a $\varrho$-Cauchy sequence in $Q_{\varrho }.$ Since $Q_{\varrho }$ is $\varrho$-complete, it is $\varrho$-convergent in $Q_{\varrho }.$ Define the mapping $W:\Lambda \rightarrow Q_{\varrho }$ by

that is,

Now, consider

for all $s_{1}\in \Lambda$ and all $\mu > 1.$ Thus, the inequality is founded without utilizing the Fatou property. Letting $\mu \rightarrow \infty,$ we have estimate of (2.17) of $\Omega$ as $W$. Replacing $\left(s_{1}, s_{2}\right)$ with $\left(\frac{s_{1}}{3^{\mu }}, \frac{s_{2}}{3^{\mu }}\right)$ in (2.16), one gets

It follows from the convexity of $\varrho$ that

for all $s_{1}, s_{2}\in \Lambda.$ Then the function $W$ is a quartic (it is enough to let $\mu \rightarrow \infty$).

For the uniqueness, suppose that $W^{\ast }:\Lambda \rightarrow Q_{\varrho }$ is a QM such that

Then, from the equations $W(3^{-\mu }s_{1}) = 3^{-4\mu }W(s_{1})$ and $W^{\ast }(3^{-\mu }s_{1}) = 3^{-4\mu }W^{\ast }(s_{1}),$ one can write

for all sufficiently large integers $\mu.$ Letting $\mu \rightarrow \infty,$ we conclude that $W\left(s_{1}\right) = W^{\ast }(s_{1})$, for all $s_{1}\in \Lambda.$ This completes the proof. □

Corollary 2.3. Suppose that $\left(\Lambda, \left\Vert.\right\Vert \right)$ is a normed space and $Q_{\varrho }$ fulfills $\Delta _{2}$-condition. Assume also there are $\xi > 0$, $p > \log _{3}\frac{u^{4}}{3}$ and the mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that

Then there exists a unique QM $W:\Lambda \rightarrow Q_{\varrho }$ fulfilling

Without utilizing the $\Delta _{2}$-condition or the Fatou property, we provide another stability result in an MS. □

Theorem 2.4. Assume that there are a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ that fulfills (2.16) and a function $\Xi :\Lambda \times \Lambda \rightarrow \lbrack 0, \infty)$ such that

Then there exists a unique QM $W:\Lambda \rightarrow Q_{\varrho }$ fulfilling

Proof. Setting $s_{2} = 0$ in (2.16), one can write

Taking $\Omega ^{\prime }\left(s_{1}\right) = \Omega \left(s_{1}\right) - \frac{1}{4}\Omega \left(0\right),$ then by the convexity of $\varrho$ and using the fact $\sum\limits_{j = 0}^{\mu -1}\frac{1}{3^{4(j+1)}} < 1,$ we get

Then one gets $\left\{ 3^{4\mu }\Omega ^{\prime }\left(\frac{s_{1}}{3^{\mu }}\right) \right\}$ is a $\varrho$-Cauchy sequence and the mapping $W:\Lambda \rightarrow Q_{\varrho }$ is defined as

that is

without utilizing the $\Delta _{2}$-condition and the Fatou property. Furthermore, it is clear from the proof used in Theorem 2.3 that the mapping $W$ satisfies the quartic functional equation.

Now, using the Fatou property and the $\Delta _{2}$-condition, we demonstrate that (2.19) is true. According to the convexity of $\varrho$ and using the fact $\sum\limits_{j = 0}^{\mu -1}\frac{1}{3^{4(j+1)}} +\frac{1}{3^{4}} < 1,$ we have

for all $s_{1}\in \Lambda$ and all natural number $\mu > 1.$ Letting $\mu \rightarrow \infty$ in the above inequality, we get our desired result.

Corollary 2.4. Assume that there exists a function $\Xi :\Lambda \times \Lambda \rightarrow \lbrack 0, \infty)$ such that

where $M\in (0, 1).$ If there exists a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ fulfilling (2.16), then there exists a unique QM $W:\Lambda \rightarrow Q_{\varrho }$ fulfilling

Corollary 2.5. Let $\left(\Lambda, \left\Vert.\right\Vert \right)$ be a normed linear space. If there are the real numbers $\xi > 0,$ $\varepsilon > 0$ and a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that

for all $s_{1}, s_{2}\in \Lambda,$ then there exists a unique QM $W:\Lambda \rightarrow Q_{\varrho }$ fulfilling

where $s_{1}\neq 0$ if $p < 0.$

2.3.   Sextic functional equation and stability study

Here, without using the Fatou property, the refined Ulam stability of the following sextic functional equation is introduced:

in an MS $Q_{\varrho }.$

Theorem 2.5. Let $Q$ be a linear space and $Q_{\varrho }$ fulfilling the $\Delta _{2}$-condition with a mapping $\Xi :\Lambda \times \Lambda \rightarrow \lbrack 0, \infty).$ Assume also there is a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that

and

for all $s_{1}, s_{2}\in \Lambda.$ Then there exists a unique sextic mapping (SM) $W:\Lambda \rightarrow Q_{\varrho },$ described as

and

for all $s_{1}\in \Lambda.$

Proof. Firstly, let $\Omega (0) = 0$ in view of $\Xi \left(0, 0\right) = 0$ along the convergence of

Setting $s_{1} = s_{2}$ in (2.20), we get

Since $\sum\limits_{j = 1}^{\infty }\frac{1}{2^{j}} < 1,$ and using $\Delta _{2}$-condition of $\varrho,$ the next functional inequality can be written as

Replacing $s_{1}$ with $\frac{s_{1}}{2^{\mu }}$ in (2.22), we see that the series in (2.16) converges, and

for all $s_{1}\in \Lambda,$ which goes to $0$ as $\theta \rightarrow \infty$ since $\frac{2}{u}\leq 1,$ then the right-hand side of the above inequality tends to $0.$ This proves that $\left\{ 2^{6\mu }\Omega \left(\frac{s_{1}}{2^{\mu }}\right) \right\}$ is a $\varrho$-Cauchy sequence for all $s_{1}\in \Lambda \ $and it is $\varrho$-convergent in $Q_{\varrho }.$ Hence, we can define the mapping $W:\Lambda \rightarrow Q_{\varrho }$ by

for all $s_{1}\in \Lambda.$ Consequently, without employing the Fatou property from the $\Delta _{2}$-condition, the following inequality

is true for all $s_{1}\in \Lambda$ and a nature number $\mu > 1.$ Letting $\mu \rightarrow \infty,$ we have estimate of (2.17) in $\Omega$ by $W$. Replacing $\left(s_{1}, s_{2}\right)$ with $\left(\frac{s_{1}}{2^{\mu } }, \frac{s_{2}}{2^{\mu }}\right)$ in (2.21), we obtain that

It follows from the convexity of $\varrho$ that

for all $s_{1}, s_{2}\in \Lambda.$ Therefore, the mapping $W$ is sextic (it is enough to let $\mu \rightarrow \infty$).

For the uniqueness, let $W^{^{\prime \prime }}:\Lambda \rightarrow Q_{\varrho }$ be another SM satisfying

From the equations $W(2^{-\mu }s_{1}) = 2^{-6\mu }W(s_{1})$ and $W^{^{\prime \prime }}(2^{-\mu }s_{1}) = 2^{-6\mu }W^{^{\prime \prime }}(s_{1}),$ we have

for all $s_{1}\in \Lambda$ and for all sufficiently large natural numbers $\mu$. Letting $\mu \rightarrow \infty,$ we obtain that $W = W^{^{^{\prime \prime }}}$ and this completes the proof. □

Corollary 2.6. Let $\left(\Lambda, \left\Vert.\right\Vert \right)$ be a normed space and $Q_{\varrho }$ fulfill the $\Delta _{2}$-condition. If there exist a real number $\xi > 0$, $p > \log _{2}\frac{u^{6}}{2}$ and a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that

for all $s_{1}, s_{2}\in \Lambda,$ then there exists a unique SM $W:\Lambda \rightarrow Q_{\varrho }$ fulfilling

Now, without employing the $\Delta _{2}$-condition and the Fatou property, the following theorem provides an alternative stability result of Theorem 2.5 in an MS.

Theorem 2.6. If there exist a mapping $\Omega :\Lambda \rightarrow Q_{\varrho },$ which fulfills (2.20) with the function $\Xi :\Lambda \times \Lambda \rightarrow \lbrack 0, \infty)$ satisfying

then there exists a unique SM $W:\Lambda \rightarrow Q_{\varrho }$ such that

Proof. Putting $s_{1} = s_{2}$ in (2.20), we have

where $\widehat{\Omega }\left(s_{1}\right) = \Omega \left(s_{1}\right) - \frac{1}{7}\Omega \left(0\right).$ From the convexity of $\varrho$ and using the fact $\sum\limits_{j = 0}^{\mu -1}\frac{1}{2^{6(j+1)}} < 1,$ we get

It follows that the sequence $\left\{ \frac{\widehat{\Omega }\left(2^{\mu }s_{1}\right) }{2^{6\mu }}\right\}$ is $\varrho$-Cauchy and the mapping $W:\Lambda \rightarrow Q_{\varrho }$ is defined by

that is,

without utilizing the $\Delta _{2}$-condition and the Fatou property. Clearly, from the proof of Theorem 2.5, we conclude that the mapping $W$ satisfies the sextic functional equation.

Now, using the Fatou property and the $\Delta _{2}$-condition, we show that (2.23) holds. From the convexity property of $\varrho$ and since $\sum\limits_{j = 0}^{\mu -1}\frac{1}{2^{6(j+1)}}+\frac{1}{2^{6}} < 1,$ we get

for all $s_{1}\in \Lambda$ and all natural number $\mu > 1.$ As $\mu \rightarrow \infty$ in the above inequality, we obtain our needed result. □

Corollary 2.7. Assume that there exists a mapping $\Xi :\Lambda \times \Lambda \rightarrow \lbrack 0, \infty)$ satisfying

for all $s_{1}, s_{2}\in \Lambda$ and for some $M^{\ast }\in (0, 1).$ If there exists a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ fulfilling (2.20), then there exists a unique SM $W:\Lambda \rightarrow Q_{\varrho }$ verifying

Corollary 2.8. Let $\left(\Lambda, \left\Vert.\right\Vert \right)$ be a normed space. Assume that there are $\xi > 0,$ $\varepsilon > 0,$ $p\in (-\infty, 2)$ and a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that

for all $s_{1}, s_{2}\in \Lambda.$ Then there exists a unique SM $W:\Lambda \rightarrow Q_{\varrho }$ fulfilling

for all $s_{1}\in \Lambda.$

3.   Stability analysis in 2-Banach spaces

In this section, we discuss the stability of the involved functional equations by considering $\Lambda$ as a linear normed space and $Q$ as a $2$-BS.

3.1.   Stability of Cauchy additive functional equation

For the convenience of notations, define the mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ as

for each $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda.$

Theorem 3.1. Suppose that there exists a function $\Xi :\Lambda ^{4}\times Q\rightarrow \lbrack 0, \infty)$ such that

If there is a mapping $\Omega :\Lambda \rightarrow Q$ with $\Omega (0) = 0$ such that

and

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda$ and $\ell \in Q,$ then there is a unique AM $V:\Lambda \rightarrow Q$ fulfilling

Proof. Setting $s_{1} = s_{2} = s_{3} = s_{4}$ in (3.2), we get

Replacing $s_{1}$ with $4^{j}s_{1}$ in (3.4), and using

for all $s_{1}\in \Lambda,$ $\ell \in Q$ and all $j > 0$, one writes

for all $s_{1}\in \Lambda,$ $\ell \in Q$ and all integers $j > 0$ and $r > 0$ with $r\leq j.$ It follows from (3.4) and (3.5) that the sequence $\left\{ \frac{\Omega \left(4^{j}s_{1}\right) }{4^{j}}\right\}$ is a Cauchy sequence in $Q$. The completeness of $Q$ implies that the sequence $\left\{ \frac{\Omega \left(4^{j}s_{1}\right) }{4^{j}}\right\}$ converges in $Q$ for all $s_{1}\in \Lambda.$ Therefore, we can define that mapping $V:\Lambda \rightarrow Q$ as

Hence,

Putting $t = 0$ and let $j\rightarrow \infty$ in (3.5), we have (3.3).

Now, we shall show that $V$ is an AM. Using (3.1), (3.2), (3.6) and Lemma 1.2, one gets

By Lemma 1.1,

Thus, $V$ is an AM. For the uniqueness, consider another AM $V^{\prime }:\Lambda \rightarrow Q$ fulfilling (3.3). Then,

Based on Lemma 1.1, we have $V(s_{1})-V^{\prime }(s_{1}) = 0$ for all $s_{1}\in \Lambda,$ which implies that $V = V^{\prime }.$ □

Corollary 3.1. Assume that $\mu :[0, \infty)\rightarrow \lbrack 0, \infty)$ is a function such that $\mu (0) = 0$ and the following assertions hold:

$(a_{1})$ $\mu (\varkappa \omega)\leq \mu (\varkappa)\mu (\omega);$

$(a_{2})$ For all $\varkappa > 1,$ $\mu (\varkappa) < \varkappa.$

If there exists a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ such that $\Omega (0) = 0$ and

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda$ and $\ell \in Q,$ then there is a unique AM $V:\Lambda \rightarrow Q$ fulfilling

Proof. Consider

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda$ and $\ell \in Q.$ Based on condition $(a_{1}),$ we have

and

Using Theorem 3.1, we obtain (3.7). □

Corollary 3.2. Assume that $\Game :\left([0, \infty)\right) ^{4}\rightarrow \lbrack 0, \infty)$ is a homogeneous function with degree $q$ and $\Omega :\Lambda \rightarrow Q$ is a mapping satisfying $\Omega (0) = 0$ and

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda$ and $\ell \in Q.$ Then there is a unique AM $V:\Lambda \rightarrow Q$ fulfilling

for all $s_{1}\in \Lambda$ and all $\ell \in Q,$ where $p\in \mathbb{R} ^{+}$ with $p < 1.$

Corollary 3.3. Let $c\in \mathbb{R} ^{+}$ with $c < 1$ and $\Game :\left([0, \infty)\right) ^{4}\rightarrow \lbrack 0, \infty)$ be a homogeneous function with degree $a.$ Assume that $\Omega :\Lambda \rightarrow Q$ is a mapping satisfying $\Omega (0) = 0$ and

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda$ and $\ell \in Q.$ Then there is a unique AM $V:\Lambda \rightarrow Q$ fulfilling

for all $s_{1}\in \Lambda$ and all $\ell \in Q.$

Corollary 3.4. Assume that a mapping $\Omega :\Lambda \rightarrow Q$ satisfies $\Omega (0) = 0$ and

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \Lambda$ and $\ell \in Q.$ Then there is a unique AM $V:\Lambda \rightarrow Q$ fulfilling

for all $s_{1}\in \Lambda$ and all $\ell \in Q,$ where $b\in \mathbb{R} ^{+}$ with $b < 1.$

3.2.   Stability of the quartic functional equation

In this part, we assume that the mapping $\Omega :\Lambda \rightarrow Q$ is described as

for all $s_{1}, s_{2}\in \Lambda.$

Theorem 3.2. Assume that $\Xi :\Lambda ^{4}\times Q\rightarrow \lbrack 0, \infty)$ is a function such that

for all $s_{1}, s_{2}\in \Lambda$ and $\ell \in Q.$ If there exists $\Omega :\Lambda \rightarrow Q$ with $\Omega (0) = 0$ such that

and

for all $s_{1}\in \Lambda$ and $\ell \in Q,$ then there exists a unique QM $V_{4}:\Lambda \rightarrow Q$ fulfilling

Proof. Consider $s_{2} = 0$ in (3.9). We have

which implies that

for all $s_{1}\in \Lambda$ and $\ell \in Q.$ In (3.11), replace $s_{1}$ with $3^{j}s_{1},$ to get

for all $s_{1}\in \Lambda$, $\ell \in Q$ and all integer $j > 0$. Hence,

for all $s_{1}\in \Lambda$, $\ell \in Q$ and all integers $j\geq m > 0$. Therefore, from (3.9) and (3.12), the sequence $\left\{ \frac{\Omega \left(3^{j}s_{1}\right) }{3^{4j}}\right\}$ is a Cauchy sequence in $Q$. The completeness of $Q$ implies that the sequence $\left\{ \frac{\Omega \left(3^{j}s_{1}\right) }{3^{j}}\right\}$ converges in $Q$ for all $s_{1}\in \Lambda.$ Therefore, we can describe the mapping $V_{4}:\Lambda \rightarrow Q$ as

Hence,

Letting $m = 0$ and $j\rightarrow \infty$ in (3.12), we have (3.10).

Now, we shall show that $V_{4}$ is a QM. Using (3.8), (3.9), (3.13) and Lemma 1.2, one can write

for all $s_{1}\in \Lambda$, $\ell \in Q.$ By Lemma 1.1, we get

Thus, $V$ is a QM. For the uniqueness, consider another QM $V_{4}^{\prime }:\Lambda \rightarrow Q$ fulfilling (3.10). Then

Based on Lemma 1.1, we have $V_{4}(s_{1})-V_{4}^{\prime }(s_{1}) = 0$ for all $s_{1}\in \Lambda,$ which implies that $V_{4} = V_{4}^{\prime }$ and this completes the proof. □

Corollary 3.5. Let $\mu :[0, \infty)\rightarrow \lbrack 0, \infty)$ be a given function with $\mu (0) = 0$ and

$(i)$ $\mu (\varkappa \omega)\leq \mu (\varkappa)\mu (\omega),$

$(ii)$ for all $\varkappa > 1,$ $\mu (\varkappa) < \varkappa.$

If there exists a mapping $\Omega :\Lambda \rightarrow Q_{\varrho }$ with $\Omega (0) = 0$ and

for all $s_{1}, s_{2}\in \Lambda$ and $\ell \in Q,$ then there exists a unique QM $V_{4}:\Lambda \rightarrow Q$ fulfilling

Proof. Assume that

for all $s_{1}, s_{2}\in \Lambda$ and $\ell \in Q.$ From the condition $(i),$ we get

and

By utilizing Theorem 3.1, we obtain (3.14). □

Corollary 3.6. Assume that $\Game :\left([0, \infty)\right) ^{2}\rightarrow \lbrack 0, \infty)$ is a homogeneous function with degree $q$ and $\Omega :\Lambda \rightarrow Q$ is a mapping satisfying $\Omega (0) = 0$ and

for all $s_{1}, s_{2}\in \Lambda$ and $\ell \in Q.$ Then there exists a unique QM $V_{4}:\Lambda \rightarrow Q$ fulfilling

for all $s_{1}\in \Lambda$ and all $\ell \in Q,$ where $p\in \mathbb{R} ^{+}$ with $p < 1.$

Corollary 3.7. Let $c\in \mathbb{R} ^{+}$ with $c < 1$ and $\Game :\left([0, \infty)\right) ^{4}\rightarrow \lbrack 0, \infty)$ be a homogeneous function with degree $a.$ Assume that $\Omega :\Lambda \rightarrow Q$ is a mapping satisfying $\Omega (0) = 0$ and

for all $s_{1}, s_{2}\in \Lambda$ and $\ell \in Q.$ Then there exists a unique AM $V_{4}:\Lambda \rightarrow Q$ fulfilling

for all $s_{1}\in \Lambda$ and all $\ell \in Q.$

Proof. We obtain the proof immediately, if we take in Theorem 3.2,

for all $s_{1}, s_{2}\in \Lambda$ and $\ell \in Q$. □

Corollary 3.8. Assume that a mapping $\Omega :\Lambda \rightarrow Q$ satisfies $\Omega (0) = 0$ and

for all $s_{1}, s_{2}\in \Lambda$ and $\ell \in Q.$ Then there exists a unique AM $V:\Lambda \rightarrow Q$ fulfilling

for all $s_{1}\in \Lambda$ and all $\ell \in Q,$ where $b\in \mathbb{R} ^{+}$ with $b < 1.$

3.3.   Stability results of the sextic functional equation

By the same methods used in sections 3.1 and 3.2, we can obtain the refined stability of the sextic functional equation by defining a mapping $\Omega :\Lambda \rightarrow Q$ as

for all $s_{1}, s_{2}\in \Lambda,$ where $\Lambda$ is a linear normed spaces and $Q$ is a $2$-BS.

4.   Counter-examples

With the aid of a pertinent example, it is demonstrated that the functional equations (2.1) and (2.15) are unstable in the singular condition. To Gajda's outstanding example in ^[30], which demonstrates the instability in Corollaries 2 and 3 of equations (2.1) and (2.15), respectively, we propose the following examples as counter-examples via the assumptions $p\neq 1$ and $p\neq \log _{3}\frac{ u^{4}}{3},$ respectively.

Here, $\mathbb{R}$ stands for a real space, $\mathbb{Z}$ and $\mathbb{Q}$ refer to the sets of integer ad rational numbers. Our counter-examples can be demonstrated as in ^[31,32].

Remark 4.1. If a mapping $\Omega : \mathbb{R} \rightarrow \Lambda$ fulfills the functional equation (2.1), then the following assertions are true:

$(R_{1})$ For all $s_{1}\in \mathbb{R},$ $z\in \mathbb{Z}$ and $m\in \mathbb{Q},$ $\Omega \left(m^{z}s_{1}\right) = m^{z}\Omega \left(s_{1}\right);$

$(R_{2})$ For all $s_{1}\in \mathbb{R},$ if the mapping $\Omega$ is continuous, then $\Omega \left(s_{1}\right) = s_{1}\Omega \left(1\right).$

Example 4.1. Assume that $\Omega : \mathbb{R} \rightarrow \mathbb{R}$ is a function described as

where

If we define a function $\Omega : \mathbb{R} \rightarrow \mathbb{R}$ as in (2.1) such that

for all $s_{1}, s_{2}, s_{3}, s_{4}\in \mathbb{R},$ then we cannot found an AM $W: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies

for all $s_{1}\in \mathbb{R},$ where $\xi$ and $\varkappa$ are constants.

Remark 4.2. If a mapping $\Omega : \mathbb{R} \rightarrow \Lambda$ fulfills the functional equation (2.15), then the following hypotheses hold:

$(R_{1})$ For all $s_{1}\in \mathbb{R},$ $z\in \mathbb{Z}$ and $m\in \mathbb{Q},$ $\Omega \left(m^{\frac{z}{3}}s_{1}\right) = m^{z}\Omega \left(s_{1}\right);$

$(R_{2})$ For all $s_{1}\in \mathbb{R},$ if a mapping $\Omega$ is continuous, then $\Omega \left(s_{1}\right) = s_{1}^{3}\Omega \left(1\right).$

Example 4.2. Assume that $\Omega : \mathbb{R} \rightarrow \mathbb{R}$ is a function described as

where

If we define a function $\Omega : \mathbb{R} \rightarrow \mathbb{R}$ as in (2.15) such that

for all $s_{1}, s_{2}\in \mathbb{R},$ then we cannot found a QM $W: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies

for all $s_{1}\in \mathbb{R},$ where $\xi$ and $\varkappa$ are constants.

5.   Conclusions and future works

The concept of stability of a functional equation arises if we replace this functional equation by an inequality acting as a perturbation on the equation itself. Stability of the functional equation has been become an interesting subject over the last seventy years. Several results appeared in this direction. In our work, the direct method of Hyers has been utilized to study the refined stability of the additive, quartic, and sextic functional equations in modular spaces with and without the $\Delta _{2}$-condition. Moreover, we used the direct approach to investigate the Ulam stability in $2$-Banach spaces. At the end, some counter examples have been presented in order to ensure that the stability of these equations does not hold in a particular case. As future works, we look forward to study the stability of generalized additive, generalized quartic and generalized sextic functional equations. We will also study the effect of the multivalued mappings of these equations on $D$-metric spaces and generalized metric spaces.

Acknowledgments

This work was supported in part by the Basque Government under Grant IT1555-22.

Conflict of interest

The authors declare that they have no competing interests.