A new approach for Cauchy noise removal

1.   Introduction

The stability problem for the functional equation was conjectured by Ulam ^[1] in 1940. In the following year, Hyers ^[2] presented a partial answer for the case of the additive mapping in this problem: If $f$ satisfies $|f(x+y)-f(x)-f(y)|\leq \varepsilon$ for some fixed $\varepsilon > 0$, then there is an additive mapping $g$ satisfying $g(x+y) = g(x)+g(y)$ and $|f(x)-g(x)|\le \varepsilon$, which is called the Hyers-Ulam stability.

Baker et al. ^[3] announced, in 1979, the new concept for the superstability as follows: If $f$ satisfies $|f(x+y)-f(x)f(y)|\leq \varepsilon$ for some fixed $\varepsilon > 0$, then either $f$ is bounded or $f$ satisfies the exponential functional equation $f(x+y) = f(x)f(y).$

Baker ^[4] showed the superstability of the cosine (also called d'Alembert) functional equation

The cosine (d'Alembert) functional equation (C) was generalized to the following:

in which ($W$) is called the Wilson equation, ($K_{gf}$) is called the Kim's equation and remaining equations are raised in Kim's papers ^[5,6,7].

The superstability of the trigonometric (cosine (C), Wilson ($W$), Kim (($K_{gf}$), (T), ($T_{gf}$) and ($T_{gh}$)) functional equations were founded in Badora ^[8], Badora and Ger ^[9], Kannappan and Kim ^[10], Kim and Dragomir ^[11], Kim ^[5,6,7] and in papers ^[13,14,15].

In 1983, Cholewa ^[16] proved the superstability of the sine functional equation

under the stability inequality bounded by constant. This was improved to the condition bounded by a function in Badora and Ger ^[9]. Their results were also further improved later by Kim ^[17,18], who obtained the superstability under the assumption that the stability inequality is bounded by a constant or a function for the generalized sine functional equations:

In 2009, Eshaghi Gordji and Parviz ^[19] introduced the quadratic-radical functional equation

related to the additive mapping and proved its stability.

Recently, Almahalebi et al. ^[20], Kim ^[21,22] obtained the superstability of $p$-radical functional equations in relation with Wilson ($W$), Kim (($K_{gf}$), ($T_{gf}$) and ($T_{gh}$)). In the concept of the $p$-radical, the sine functional equation ($S$) and (S)-type's equations ($S_{fg}$), ($S_{gf}$), ($S_{gg}$), ($S_{gh}$) are expressed as follows:

In the above, letting $f(x) = F(x^p)$, then $F$ satisfies (S)-type equations. Hence, in this paper, they will be reasonably called the $p$-power-radical equation. Since the function $f(x) = \sin x^p$ is the solution of the equation ($S^{r}$), it will be called the $p$-power-radical sine functional equation.

Our aim of this paper is to find solutions and to investigate the superstability bounded by a function (Gǎvruta sense) for the $p$-power-radical sine functional equation ($S^{r}$) from an approximation of the $p$-power-radical functional equation ($S^{r}_{gh}$).

As a corollary, we obtain the superstability bounded by a constant and a function for the $p$-power-radical sine functional equation ($S^{r}$) from an approximation of the $p$-power-radical functional equations ($S^{r}$), ($S^{r}_{gf}$), ($S^{r}_{fg}$), ($S^{r}_{gg}$). Moreover, the obtained results are extended to Banach algebras.

In this paper, let $\mathbb{R}$ be the field of real numbers, $\mathbb{R}_{+} = [0, \infty)$ and $\mathbb{C}$ be the field of complex numbers. We assume that $f, g, h$ are nonzero functions, $\varepsilon$ is a nonnegative real number, $\varphi : \mathbb{R} \rightarrow \mathbb{\mathbb{R}_{+}}$ is a given nonnegative function and $p$ is a positive odd integer.

2.   Solutions of the functional equations

Let's recall the trigonometric formula, the $p$-power-radical functional equation's forms for the functional equations (cosine (d'Alembert) (C), Wilson ($W$), Kim ($T_{gf}$), ($T_{gg}$) and ($T_{gh}$) are the following:

We can confirm that each equation has a solution as follows: $( C^{r} ): f(x) = \cos(x^{p})$, $( W^{r} ): f(x) = \sin(x^{p}), g(x) = \cos(x^{p})$,$( T_{gf}^{r} ): f(x) = sin(x^{p})$, $g(x) = \cos(x^{p}), ( T_{gg}^{r} ): f(x) = \cos(x^{p})$, $g(x) = i\sin(x^{p}), ( T_{gh}^{r} ): f(x) = \cos(x^{p})$, $g(x) = \sin(x^{p}), h(x) = -\sin(x^{p})$ .

In addition, the solution of each equation can also be found in perspective of the hyperbolic function, exponential function and $p$-power function, simultaneously.

Letting p = 1 in the above paragraph, we know that each original equation ((C), ($W$), ($T_{gf}$), ($T_{gg}$), ($T_{gh}$)) has the corresponding solution of the same form, respectively. They also are represented by the hyperbolic function, exponential function and $p$-power function, simultaneously.

Now let's consider the functional equations generated by the product of the above equations, then we obtain the target equations: $p$-power-radical functional equation ($S^{r}$) and ($S^r$)-type's equations ($S^{r}_{fg}$), ($S^{r}_{gf}$), ($S^{r}_{gg}$) and ($S^{r}_{gh}$).

1) ($S^{r}$) has a solution as the $p$-power function $f(x) = x^p$:

2) When ($S^{r}$) has a solution as the sine function, it also has simultaneously as an exponential solution as follows:

3) When ($S^{r}$) has a solution as the hyperbolic sine function, it also has simultaneously as an exponential solution as follows:

Although the mentioned all functional equations may have arisen from sine or cosine, as shown in the previous, they have solutions as the $p$-power, the hyperbolic and the exponential function, simultaneously. Hence, they can be considered as the $p$-power-radical, the $p$-power-radical exponential and the $p$-power-radical hyperbolic functional equation, simultaneously.

Letting p = 1 in the above items 1), 2) and 3), then ($S^{r}$) arrives ($S$). Hence, based on the above reasons, the Eq ($S$) well-known as the sine function equation can also be called as the $p$-power, the exponential and the hyperbolic functional equation, simultaneously.

In the following lemma, we find the forms of solutions of the $p$-power-radical functional equations ($S^{r}_{gh}$), ($S^{r}_{gg}$), ($S^{r}_{fg}$).

Lemma 1. If $f, g, h: \mathbb{R} \rightarrow \mathbb{C}$ satisfy ($S^{r}_{gh}$), then, as one of the solutions of ($S^{r}_{gh}$), $f, g, h$ have the forms $f(x) = \cos(x^p)$, $g(x) = \sin(x^p)$ and $h(x) = -\sin(x^p)$ for all $x \in \mathbb{R}$.

Proof. For all $x, y \in \mathbb{R}$,

In the next lemma, let's find an exponential solution for ($S^{r}_{fg}$).

Lemma 2. If $f, g: \mathbb{R} \rightarrow \mathbb{C}$ satisfy ($S^{r}_{fg}$), then, as the solutions of ($S^{r}_{fg}$), $f, g$ have the following two forms

(i) $f(x) = e^{{x}^p}$, $g(x) = e^{{x}^p}-e^{-{x}^p}$ for all $x \in \mathbb{R}$,

(ii) $f(x) = e^{{x}^p}$, $g(x) = 2 \sinh(x^p)$ for all $x \in \mathbb{R}$.

Proof. For all $x, y \in \mathbb{R}$,

In the next lemma, let's find a hyperbolic and trigonometric solution for ($S^{r}_{gg}$).

Lemma 3. If $f, g: \mathbb{R} \rightarrow \mathbb{C}$ satisfy ($S^{r}_{gg}$), then, as the solutions of ($S^{r}_{gg}$), $f, g$ have the following two forms

(i) $f(x) = \cosh(x^{p})$, $g(x) = \sinh(x^{p})$,

(ii) $f(x) = \cos(x^p)$, $g(x) = i \sin(x^p)$.

Proof. For all $x, y \in \mathbb{R}$,

3.   Superstability of ($S^{r}$) from ($S^{r}_{gh}$)

In Section 3, we study the superstability of the $p$-power-radical sine functional equation ($S^{r}$) from an approximation of the $p$-power-radical functional equation ($S^{r}_{gh}$) related to ($S$).

Theorem 1. Assume that $f, g, h: \mathbb{R} \longrightarrow \mathbb{C}$ satisfy the inequality

for all $x, y\in \mathbb{R}$, where $p$ is a positive odd integer.

Then, either $g$ is bounded or $h$ satisfies ($S^{r}$). Moreover, if $g$ satisfies ($C^{r}$), then $h$ and $g$ satisfy $p$-power-radical equation ($T_{gf}^{r}$): = $h(\sqrt[p]{x^{p}+y^{p}})-h(\sqrt[p]{x^{p}-y^{p}}) = 2g(x)h(y)$.

Proof. By putting $x = \sqrt[p]{2}x$ and $y = \sqrt[p]{2}y$ in (3.1), it is written equivalently as

Assume that $g$ is unbounded. Then, we can choose a sequence $\{x_{n}\}$ in $\mathbb{R}$ such that

Taking $x = x_{n}$ in (3.2), we get

and by (3.3), we get

Replacing $x$ by $\sqrt[p]{2x_{n}^{p}+x^{p}}$ and $\sqrt[p]{2x_{n}^{p}-x^{p}}$ in (3.1), we have

for all $x, y\in \mathbb{R}$ and $n\in \mathbb{N}$. Consequently,

for all $x, y\in \mathbb{R}$ and $n\in \mathbb{N}$. Taking $n\rightarrow \infty$ in (3.6) and using (3.3) and (3.4), we reach a conclusion that, for every $x\in \mathbb{R}$, there exists the limit function

where $L_{1}: \mathbb{R}\rightarrow \mathbb{C}$ satisfies the equation as even

From the definition of $L_{1}$, we obtain the equality $L_{1}(0) = 2$, which, jointly with (3.7), indicates that $h$ is odd. Keeping this in mind, through (3.7), we deduce the equality

The oddness of $h$ imposes it to vanish at $0$. Putting $x = y$ in (3.7), we conclude with the previous result that

The (3.8) by (3.9) arrives to the equation

for all $x, y\in \mathbb{R}$, which, with $\sqrt[p]{2}$-divisibility of $\mathbb{R}$, states conclusively ($S^{r}$).

In addition, if $g$ satisfies ($C^{r}$) and $L_{1}$ forces $2g$, then (3.7) forces that $h$ and $g$ satisfy ($T_{gf}^{r}$).

Theorem 2. Suppose that $f, g, h: \mathbb{R} \longrightarrow \mathbb{C}$ satisfy

which satisfies one of two cases $g(0) = 0$, $f(x)^{2} = f(-x)^{2}$, where $p$ is a positive odd integer.

Then, either $h$ is bounded or $g$ satisfies ($S^{r}$). In addition, if $h$ satisfies ($C^{r}$), then $g$ and $h$ satisfy the $p$-power-radical Wilson type equation ($W^{r}$): = $g\left(\sqrt[p]{x^{p}+y^{p}}\right)+g\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)h(y)$.

Proof. Let $h$ be unbounded, then we can select a sequence $\{y_n\}$ in $\mathbb{R}$ such that $h(\sqrt[p]{2}y_n)|\rightarrow \infty$ as $n\rightarrow \infty$. With a minor change of the steps shown in the start part of the proof in Theorem 1, we can get

Replacing $y$ by $\sqrt[p]{y^{p}+2{y_{n}^{p}}}$ and $\sqrt[p]{-y^{p}+2y_{n}^{p}}$ in (3.10), the same procedure of (3.5) and (3.6) allows, with (3.11), use to argue the existence of a limit function

where $L_{2}: \mathbb{R} \rightarrow \mathbb{C}$ satisfies the equation

Hence, from the definition of $L_{2}$, $L_{2}$ is even and $L_{2}(0) = 2$.

Let's start with the case $g(0) = 0$. Then it leads to the conclusion, by (3.12), that $g$ is odd.

Putting $y = x$ in (3.12), we obtain

From (3.12), the oddness of $g$ and (3.13), we obtain the equation

for all $x, y \in \mathbb{R}$, which, with $\sqrt[p]{2}$-divisibility of $\mathbb{R}$, states conclusively ($S^{r}$).

Second, let's consider the case $f(x)^{2} = f(-x)^{2}$. in this case, it is sufficient to show that $g(0) = 0$.

Suppose that it is not the case. Then, without loss of generality, we may assume in following that $g(0) = c$ (constant).

Taking $x = 0$ in (3.10), from the above assumption, we get the inequality

The above inequality indicates that $h$ is globally bounded, which is a contradiction due to the assumption of unboundedness. Therefore the claimed $g(0) = 0$ holds, and the proof of the theorem is completed.

In addition, if $h$ satisfies ($C^{r}$) and $L_{2}$ forces $2h$, then (3.12) forces that $g$ and $h$ satisfy ($W^{r}$): = $g(\sqrt[p]{x^{p}+y^{p}})+g(\sqrt[p]{x^{p}-y^{p}}) = 2g(x)h(y)$.

The following corollary follows from Theorems 1 and 2, immediately.

Corollary 1. Assume that $f, g, h:\mathbb{R} \rightarrow \mathbb{C}$ satisfy the inequality

where $p$ is a positive odd integer.

Then

(i) either $g$ is bounded or $h$ satisfies ($S^{r}$). Moreover, if $g$ satisfies ($C^{r}$), then $h$ and $g$ satisfy ($T_{gf}^{r}$): = $h(\sqrt[p]{x^{p}+y^{p}})-h(\sqrt[p]{x^{p}-y^{p}}) = g(x)h(y)$.

(ii) either $h$ is bounded, or g satisfies ($S^{r}$) under $g(0) = 0$ or $f(x)^{2} = f(-x)^{2}$. Moreover, if $h$ satisfies ($C^{r}$), then $g$ and $h$ satisfy the Wilson equation ($W^{r}$): = $g(\sqrt[p]{x^{p}+y^{p}})+g(\sqrt[p]{x^{p}-y^{p}}) = 2 g(x)h(y)$.

4.   Application to the superstability of the Eqs ($S^{r}$), ($S^{r}_{gg}$), ($S^{r}_{gf}$) and ($S^{r}_{fg}$)

In this section, as corollaries, we obtain the superstability of the $p$-power-radical sine functional equation ($S^{r}$) from an approximation of ($S^{r}$), and ($S^{r}_{gf}$), ($S^{r}_{fg}$) and ($S^{r}_{gg}$). Their proofs follow from Theorems 1 and 2, and Corollary 1.

4.1.   Superstability of the Eq ($S_{gg}$)

Corollary 2. Assume that $f, g:\mathbb{R} \longrightarrow \Bbb{C}$ satisfy the inequality

where $p$ is a positive odd integer.

Then, either $g$ is bounded or $g$ satisfies ($S^{r}$), respectively. In particular, the case (ii) holds under the condition $g(0) = 0$ or $f(x)^{2} = f(-x)^{2}$.

4.2.   Superstability of the Eq ($S_{gf}$)

Corollary 3. Assume that $f, g: \mathbb{R} \rightarrow \mathbb{C}$ satisfy the inequality

for all $x, y\in \mathbb{R}$, where $p$ is a positive odd integer.

Then, either $g$ is bounded or $f$ satisfy ($S^{r}$). Moreover, if $g$ satisfies ($C^{r}$), $f$ and $g$ satisfy ($T_{gf}^{r}$): = $f\left(\sqrt[p]{x^{p}+y^{p}}\right)-f\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)f(y)$.

Corollary 4. Assume that $f, g: \mathbb{R} \rightarrow \mathbb{C}$ satisfy the inequality

which satisfies one of the cases $g(0) = 0$, $f(x)^{2} = f(-x)^{2}$, where $p$ is a positive odd integer.

Then, either $f$ is bounded or $g$ satisfies ($S^{r}$). Additionally, if $f$ satisfies ($C^{r}$), $g$ and $f$ satisfy ($W^{r}$): = $g\left(\sqrt[p]{x^{p}+y^{p}}\right)+g\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)f(y)$.

Corollary 5. Assume that $f, g:\mathbb{R} \rightarrow \mathbb{C}$ satisfy the inequality

for all $x, y\in \mathbb{R}$, where $p$ is a positive odd integer.

Then

(i) either $g$ is bounded or $f$ and $g$ satisfy ($S^{r}$), respectively. Additionally, if $g$ satisfies ($C^{r}$), then $f$ and $g$ satisfy ($T_{gf}^{r}$): = $f\left(\sqrt[p]{x^{p}+y^{p}}\right)-f\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)f(y)$;

(ii) either $f$ is bounded or $g$ satisfies ($S^{r}$). Additionally, if $f$ satisfies ($C^{r}$), then $g$ and $f$ satisfy the Wilson equation ($W^{r}$): = $g\left(\sqrt[p]{x^{p}+y^{p}}\right)+g\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)f(y)$.

Proof. It is sufficient to present that either $g$ is bounded or $g$ satisfies (S). The other cases follow from Corollaries 3 and 4, immediately.

The inequality (4.3) can also be presented equivalently as

First, if $f$ is bounded, then $y_{0}\in \mathbb{R}$ can be chosen such that $f(\sqrt[p]{2}y_{0}) \neq 0$. From this $y_{0}$ and (4.4), we get

Thus, it implies that $g$ is also bounded on $\mathbb{R}$. Namely, since an unboundedness of $g$ exacts it of $f$, let run along the step of Theorem 2.

The process of Theorem 2 gives us the limit (3.11), which, since $f$ satisfies ($S^{r}$) by Theorem 1, validates

By the $\sqrt[p]{2}$-divisibility of $\mathbb{R}$, we obtain $g = f$. Thus, it is true that $g$ also satisfies ($S^{r}$).

4.3.   Superstability of the Eq ($S_{fg}$)

Corollary 6. Assume that $f, g:\mathbb{R} \longrightarrow \Bbb{C}$ satisfy the inequality

where $p$ is a positive odd integer.

Then, either $f$ is bounded or $g$ satisfies ($S^{r}$). Additionally, if $f$ satisfies ($C^{r}$), then $g$ and $f$ satisfy ($T_{gf}^{r}$): = $g\left(\sqrt[p]{x^{p}+y^{p}}\right)-g\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 f(x)g(y)$.

Corollary 7. Assume that $f, g:\mathbb{R} \longrightarrow \Bbb{C}$ satisfy the inequality

where $p$ is a positive odd integer.

Then, either $g$ is bounded or $f$ satisfies ($S^{r}$) under one condition of the cases $f(0) = 0$, $f(x)^{2} = f(-x)^{2}$. In addition, if $g$ satisfies ($C^{r}$), then $f$ and $g$ satisfy the Wilson equation ($W^{r}$): = $f\left(\sqrt[p]{x^{p}+y^{p}}\right)+f\left(\sqrt[p]{x^{p}-y^{p}}\right) = f(x)g(y)$.

Corollary 8. Assume that $f, g:\mathbb{R} \longrightarrow \mathbb{C}$ satisfy the inequality

for all $x, y\in \mathbb{R}$, where $p$ is a positive odd integer.

Then

(i) either $f$ is bounded or $g$ satisfies ($S^{r}$). In addition, if $f$ satisfies ($C^{r}$), then $g$ and $f$ satisfy ($T_{gf}^{r}$): = $g\left(\sqrt[p]{x^{p}+y^{p}}\right)-g\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 f(x)g(y)$;

(ii) either $g$ is bounded or $f$ and $g$ satisfy ($S^{r}$), respectively, under one condition of the cases $f(0) = 0$, $f(x)^{2} = f(-x)^{2}$. In addition, if $g$ satisfies ($C^{r}$), then $f$ and $g$ satisfy the Wilson equation ($W^{r}$): = $f\left(\sqrt[p]{x^{p}+y^{p}}\right)+f\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 f(x)g(y)$.

4.4.   Superstability of the $p$-power-radical sine functional equation ($S^{r}$)

As a corollary for all the obtained results, we obtain the superstability of the $p$-power-radical sine functional equation ($S^{r}$).

Corollary 9. Assume that $f: \mathbb{R} \rightarrow \mathbb{C}$ satisfies the inequality

where $p$ is a positive odd integer.

Then, either $f$ is bounded or $f$ satisfies ($S^{r}$)

Proof. Replacing the functions $g$ and $h$ in Theorems 1 and 2 by $f$, in the case (ii), the assumption $f(0) = 0$ or $f(x)^{2} = f(-x)^{2}$ can be eliminated (see [9, Theorem 5]).

Remark 1. (i) Applying $\varphi (x) = \varphi (y) = \varepsilon\;$ for all results in Sections 3 and 4, then they yield the superstability results bounded by constant (Hyers-sense).

(ii) Applying '$p = 1$' to all the $p$-power-radical functional equations ($S^{r}$), ($S^{r}_{gf}$), ($S^{r}_{fg}$), ($S^{r}_{gg}$), ($S^{r}_{gh}$) in Sections 3 and 4, then they yield the superstability results for all (S)-type functional equations: ($S$), ($S_{gf}$), ($S_{fg}$), ($S_{gg}$), ($S_{gh}$).

In addition, for all results of the (S)-types obtained above, applying again (i) $\varphi(x) = \varphi(y) = \varepsilon$, then they yield the additional results (Hyers-sense) for (S)-types.

(iii) Many results obtained for the (S)-types and $p$-power-radical ($S^{r}$)-types in (i) and (ii) are found in Cholewa ^[16], Badora ^[8], Badora and Ger ^[9], Kannappan and Kim ^[10], Kim and Dragomir ^[11], Kim ^[18,22] and in papers ^{[8,9,13,14,15]}.

5.   Extension of the stability to Banach algebras

All results in Sections 3 and 4 can be expanded to the stability on Banach algebras. The following theorem is based on Theorems 1 and 2, and Corollary 1. The remainder results also are represented as similar type as Theorem 3, respectively, their proofs will skip for the sake of brevity.

Theorem 3. Let $(E, \|\cdot\|)$ be a semisimple commutative Banach algebra. Assume that $f, g, h: \mathbb{R} \longrightarrow E$ satisfy the inequality

where $p$ is a positive odd integer.

Then, for an arbitrary linear multiplicative functional $x^{*} \in E^{*}$,

$\rm (i)$ either the superposition $x^{*} \circ g$ is bounded or $h$ satisfies ($S^{r}$), In addition, if $g$ satisfies ($C^{r}$), then $h$ and $g$ satisfy ($T_{gf}^{r}$): = $h\left(\sqrt[p]{x^{p}+y^{p}}\right)-h\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)h(y)$;

$\rm (ii)$ either the superposition $x^{*} \circ h$ under the cases $g(0) = 0$ or $f(x)^{2} = f(-x)^{2}$ is bounded or $g$ satisfies ($S^{r}$). In addition, if $h$ satisfies ($C^{r}$), then $g$ and $h$ satisfy the Wilson equation ($W^{r}$): = $g\left(\sqrt[p]{x^{p}+y^{p}}\right)+g\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)h(y)$;

$\rm (iii)$ the above $\rm (i)$ and $\rm (ii)$ hold. In addition, if the superposition $x^{*} \circ g$ is unbounded, then $g$ satisfies ($S^{r}$)

Proof. (i) Assume that (i) holds and fix arbitrarily a linear multiplicative functional $x^{\ast }\in E$. As is well known, we have $\Vert x^{\ast }\Vert = 1$, whence, for every $x, y\in \mathbb{R}$, we have

which states that the superpositions $x^{\ast }\circ g$ and $x^{\ast }\circ h$ produce a solution of stability inequality (3.1) of Theorem 1. Since, by assumption, the superposition $x^{\ast }\circ g$ is unbounded, an appeal to Theorem 1 forces that the function $x^{\ast }\circ h$ is a solution of ($S^{r}$), that is,

In other presents, by the linear multiplicativity of $x^{\ast }$, for all $x, y\in \mathbb{R}$, the difference $\mathcal{D} { S^{r}}:\mathbb{R} \times \mathbb{R} \rightarrow E$ defined by

falls into the kernel of $x^{*}$. Thus, in view of the unrestricted choice of $x^{*}$, we infer that

for all $x, y \in \mathbb{R}$. Since the space $E$ is a semisimple, $\bigcap \{\ker x^{\ast }: x^{\ast } \in E^{*}\} = 0$, which means that $h$ satisfies the claimed Eq ($S^{r}$).

In addition, if $g$ satisfies ($C^{r}$), then it is trivial that $h$ and $g$ satisfy $h\left(\sqrt[p]{x^{p}+y^{p}}\right)-h\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)h(y)$.

(ii) By assumption, the superposition $z^{\ast }\circ h$ with $g(0) = 0$ or $f(x)^{2} = f(-x)^{2}$ is unbounded, an appeal to Theorem 2 shows that the results hold.

The superposition $x^{\ast }\circ g$ satisfies (5.1), that is a solution of the Eq ($S^{r}$).

As in (i), a linear multiplicativity of $x^{\ast }$ and semisimplicity imply

which means that $g$ satisfies ($S^{r}$). In addition, if $h$ satisfies ($C^{r}$), then it is trivial that $g$ and $h$ satisfy $g\left(\sqrt[p]{x^{p}+y^{p}}\right)+g\left(\sqrt[p]{x^{p}-y^{p}}\right) = 2 g(x)h(y)$.

(iii) It follows from the above (i) and (ii), and the additional case of (iii) holds by Corollary 1.

Corollary 10. Let $(E, \|\cdot\|)$ be a semisimple commutative Banach algebra. Assume that $f, g: \mathbb{R} \longrightarrow E$ satisfy the inequality

where $p$ is a positive odd integer.

For an arbitrary linear multiplicative functional $x^{*} \in E^{*}$, either the superposition $x^{\ast }\circ g$ is bounded or $g$ satisfies ($S^{r}$). In particular, the case (ii) holds under the condition $g(0) = 0$ or $f(x)^{2} = f(-x)^{2}$.

Corollary 11. Let $(E, \|\cdot\|)$ be a semisimple commutative Banach algebra. Assume that $f, g: \mathbb{R} \longrightarrow E$ satisfy the inequality

where $p$ is a positive odd integer.

Then, for an arbitrary linear multiplicative functional $x^{*} \in E^{*}$,

$\rm (i)$ either the superposition $x^{*} \circ g$ is bounded or $f$ satisfies ($S^{r}$), In addition, if $g$ satisfies ($C^{r}$), then $f$ and $g$ satisfy ($T_{gf}^{r}$);

$\rm (ii)$ either the superposition $x^{*} \circ f$ under the cases $g(0) = 0$ or $f(x)^{2} = f(-x)^{2}$ is bounded or $g$ satisfies ($S^{r}$). In addition, if $f$ satisfies ($C^{r}$), then $g$ and $f$ satisfy the Wilson equation ($W^{r}$);

$\rm (iii)$ the above $\rm (i)$ and $\rm (ii)$ hold. Also, additionally, if the superposition $x^{*} \circ g$ is unbounded, then $g$ satisfies ($S^{r}$)

Corollary 12. Let $(E, \|\cdot\|)$ be a semisimple commutative Banach algebra. Assume that $f, g: \mathbb{R} \longrightarrow E$ satisfy the inequality

where $p$ is a positive odd integer.

Then, for an arbitrary linear multiplicative functional $x^{*} \in E^{*}$,

$\rm (i)$ either the superposition $x^{*} \circ f$ is bounded or $g$ satisfy ($S^{r}$). In addition, if $f$ satisfies ($C^{r}$), then $g$ and $f$ satisfy ($T_{gf}^{r}$);

$\rm (ii)$ either the superposition $x^{*} \circ g$ under the cases $g(0) = 0$ or $f(x)^{2} = f(-x)^{2}$ is bounded or $f$ satisfies ($S^{r}$). In addition, if $g$ satisfies ($C^{r}$), then $f$ and $g$ satisfy ($W^{r}$);

$\rm (iii)$ the above $\rm (i)$ and $\rm (ii)$ hold. Also, additionally, if the superposition $x^{*} \circ g$ is unbounded, then $g$ satisfies ($S^{r}$),

Corollary 13. Let $(E, \|\cdot\|)$ be a semisimple commutative Banach algebra. Assume that $f: \mathbb{R} \longrightarrow E$ satisfies the inequality

where $p$ is a positive odd integer.

For an arbitrary linear multiplicative functional $x^{*} \in E^{*}$, either the superposition $x^{\ast }\circ f$ is bounded or $f$ satisfies ($S^{r}$).

Remark 2. Follow (i) and (ii) of Remark 1 for all results in Section 5, namely,

(i) Apply $\varphi (x) = \varphi (y) = \varepsilon\;$ in all results.

(ii) Apply '$p = 1$' in all results. Next, apply (i) again in the results.

Then, a number of the results are found in the same papers in (iii) of Remark 1.

6.   Conclusions

In this paper, we studied solutions and creating of the $p$-power-radical functional equations arisen simultaneously from the trigonometric functions, hyperbolic function, exponential function and $p$-radical function.

We investigated the superstability bounded by a function (Gǎvruta sense) for the $p$-power-radical sine functional equation ($S^{r}$) from an approximation of the $p$-power-radical functional equations ($S^{r}_{gh}$), and ($S^{r}$), ($S^{r}_{gf}$), ($S^{r}_{fg}$), ($S^{r}_{gg}$) with $p$ is a positive odd integer. Furthermore, the obtained results extended to Banach algebras. As a result, we have improved the previous stability results for (S)-type functional equations: (S), ($S_{gf}$), ($S_{fg}$), ($S_{gg}$), ($S_{gh}$) to that of the $p$-power-radical equations: ($S^{r}$), ($S^{r}_{gf}$), ($S^{r}_{fg}$), ($S^{r}_{gg}$), ($S^{r}_{gh}$).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This study was supported by research fund from Chosun University, 2023.

Conflict of interest

The authors declare there are no conflicts of interest.