Approximation of Two Systems of Radical Functional Equations Related to Quadratic and Quartic Mappings

Abstract

In this work, we define the multi-radical quadratic and multi-radical quartic mappings as two systems of radical functional equations and then represent them as two equations. Then, we establish some results concerning the stability of multi-radical quadratic and multi-radical quartic mappings by applying a fixed-point based on Brzdȩk. As a direct consequence, we prove the Rassias and Hyers stability of the mentioned mappings.

1. Introduction and Preliminaries

In the realm of functional equations, their solutions and stability are special and fundamental concepts. In other words, the types of stability functional equations have constitute a fascinating topic within the range of nonlinear analyses. Note that a functional equation $\mathcal{F}$ is said to be stable if any function f satisfying the equation $\mathcal{F}$ approximately must be close to an exact solution of $\mathcal{F}$. Ulam stability [1], proposed for group homomorphisms during a mathematical colloquium, resulted in the emergence of the stability problem in functional equations. Later, an affirmative reply to Ulam’s inquiry, in the context of Banach spaces, for the additive functional equation was given by Hyers [2]. Indeed, he proved the following theorem: Let $\mathbb{E}$ and $\mathbb{F}$ be Banach spaces. Assume that $\Gamma :\mathbb{E}⟶\mathbb{F}$ such that

$$\parallel \Gamma (e_1+e_2)-\Gamma \left(e_1\right)-\Gamma \left(e_2\right)\parallel \le ϵ,$$

for all $e_1,e_2\in \mathbb{E}$ and for some $ϵ\ge 0$. Then, there exists a unique additive mapping $\mathcal{A}:\mathbb{E}⟶\mathbb{F}$ such that $\parallel \Gamma \left(e\right)-\mathcal{A}\left(e\right)\parallel \le ϵ$ for all $e\in \mathbb{E}$, and if $\Gamma \left(te\right)$ is continuous in t for each fixed e, then $\Gamma$ is a linear mapping.

Subsequently, Hyers’ result was significantly generalized by Aoki [3], Th. M. Rassias [4] (where stability incorporated the sum of the powers of norms) and Găvruţa [5] (where stability was controlled by a general control function). This theory examines whether a function that approximately satisfies a certain functional equation is close to a function that exactly satisfies that equation.

Skof [6] (see also [7]) was the first mathematician who introduced and studied the quadratic functional equation

$$\begin{array}{c}\hfill Q(x+y)+Q(x-y)=2Q\left(x\right)+2Q\left(y\right).\end{array}$$

(1)

It is known that (1) is the key tool for characterizing inner product spaces; for more information we refer the reader to [8,9] and the references therein. Moreover, Lee et al. [10] defined the following quartic functional equation:

$$\begin{array}{c}\hfill Q(2x+y)+Q(2x-y)=4Q(x+y)+4Q(x-y)+24Q\left(x\right)-6Q\left(y\right).\end{array}$$

(2)

The functional equations

$$\begin{array}{c}\hfill R\left(\sqrt{x^2+y^2}\right)=R\left(x\right)+R\left(y\right)\end{array}$$

(3)

and

$$\begin{array}{c}\hfill R\left(\sqrt{x^2+y^2}\right)+R\left(\sqrt{|x^2-y^2|}\right)=2R\left(x\right)+2R\left(y\right)\end{array}$$

(4)

corresponding to a mapping $R:\mathbb{R}⟶V$, are called the radical quadratic functional equation and radical quartic functional equation, respectively, as presented in [11], where V is a vector space over $\mathbb{R}$. Let us note that the mappings $R_1,R_2:\mathbb{R}⟶V$ define, via $R_1\left(x\right)=x^2$ and $R_2\left(x\right)=x^4$, the solution of (2) and (4), respectively. The following observations were presented in [11]:

• Let V be a vector space over $\mathbb{R}$. If a mapping $\mathcal{Q}:\mathbb{R}⟶V$ satisfies functional Equation (3), then f is quadratic. In other words, Equation (1) is valid for it. Moreover,

(i)

$\mathcal{Q}\left(r^{{\displaystyle \frac{k}{2}}}x\right)=r^k\mathcal{Q}\left(x\right)$ for all $x\in \mathbb{R},r\in {\mathbb{Q}}^{+}$ and integers k;

(ii)

If $\mathcal{Q}$ is continuous, then $\mathcal{Q}\left(x\right)=x^2\mathcal{Q}\left(1\right)$ for all $x\in {\mathbb{R}}_{+}$.

• If the mentioned mapping $\mathcal{Q}$ fulfils Equation (4), then f is quartic. Indeed, we have

(i)

$\mathcal{Q}\left(r^{{\displaystyle \frac{k}{2}}}x\right)=r^{2k}\mathcal{Q}\left(x\right)$ for all $x\in \mathbb{R},r\in {\mathbb{Q}}^{+}$ and integers k;

(ii)

If $\mathcal{Q}$ is continuous, then $\mathcal{Q}\left(x\right)=x^4\mathcal{Q}\left(1\right)$ for all $x\in {\mathbb{R}}_{+}$.

In [11], the authors studied the stability of functional Equations (3) and (4) in the setting of two-normed spaces.

Recall that the radical equation

$$\begin{array}{c}\hfill R\left(\sqrt[s]{x^s+y^s}\right)=R\left(x\right)+R\left(y\right),\end{array}$$

(5)

for $s=3,4,5$, where R is a mapping from $\mathbb{R}$ into a normed vector space V, was defined and studied in [12,13,14]. In the case that $s=3,5,7$, we call it a radical cubic, quartic, or quintic functional equation, respectively. In the article in which the former is given, the authors established the stability of (5) in the setting of quasi-$\beta$-Banach spaces and 2-Banach spaces.

Let $(G,+)$ be a commutative group and W be a linear space. A multivariable mapping $\mathfrak{Q}:G^n⟶W$ is called multi-quadratic if it fulfils (1) for each variable. In other words, for each $i\in \{1,\dots ,n\}$, we have

$$\begin{gathered} \mathfrak{Q}(g_1,\dots ,g_{i-1},g_i+g_i^{\prime },g_{i+1},\dots ,g_n)+\mathfrak{Q}(g_1,\dots ,g_{i-1},g_i-g_i^{\prime },g_{i+1},\dots ,g_n) \\ =2\mathfrak{Q}(g_1,\dots ,g_{i-1},g_i,g_{i+1},\dots ,g_n)+2\mathfrak{Q}(g_1,\dots ,g_{i-1},g_i^{\prime },g_{i+1},\dots ,g_n). \end{gathered}$$

Moreover, $\mathfrak{Q}$ is said to be multi-quartic if it satisfies quartic functional Equation (2) in each of its n arguments; that is

$$\begin{array}{cc}\hfill & \mathfrak{Q}(g_1,\dots ,g_{i-1},2g_i+g_i^{\prime },g_{i+1},\dots ,g_n)+\mathfrak{Q}(g_1,\dots ,g_{i-1},2g_i-g_i^{\prime },g_{i+1},\dots ,g_n)\hfill \\ \hfill & =4\mathfrak{Q}(g_1,\dots ,g_{i-1},g_i+g_i^{\prime },g_{i+1},\dots ,g_n)+4\mathfrak{Q}(g_1,\dots ,g_{i-1},g_i-g_i^{\prime },g_{i+1},\dots ,g_n)\hfill \\ \hfill & +24\mathfrak{Q}(g_1,\dots ,g_{i-1},g_i,g_{i+1},\dots ,g_n)-6\mathfrak{Q}(g_1,\dots ,g_{i-1},g_i^{\prime },g_{i+1},\dots ,g_n),\hfill \end{array}$$

for all $i\in \{1,\dots ,n\}$. For more details we refer the reader to [15]. More information about the structures, equalities, inequalities, and the stability results for single and multivariable equations, such as multi-additive, multi-quadratic and multi-quartic mappings on various Banach spaces, are available in [16,17,18,19,20,21,22,23].

It is well-known that there are two classical methods for establishing the stability of a functional equation: Hyers’ (direct) [2] and fixed-point methods. It is worth mentioning that fixed-point theorems have been considered for varied mappings and functional equations, as in [24,25]. Indeed, the study of fixed-point theory provides essential tools for solving and proving the stability problems of functional equations, which are available in many articles and books.

The stability of the radical functional equation whose solutions are monomial and were mentioned in the discussion earlier motivated us to extend and generalize radical Equations (3) and (4) to multiple variables, which has not yet worked or been studied by other authors. In other words, we define the multi-radical quadratic (resp. multi-radical quartic) mappings as a system of radical quadratic (resp. quartic) functional equations. Then, we find a necessary and sufficient condition for the mapping to be a multi-radical quadratic (resp. multi-radical quartic). The benefit of presenting such systems as a single equation is that it allows us to prove the stability of multi-radical quadratic/quartic mappings by means of only one functional inequality, and not a number of functional equations. Furthermore, we establish the Găvruţa stability for the multi-radical quadratic and multi-radical quartic mappings by applying a known fixed-point theorem. Eventually, we prove the Hyers and Rassias stabilities of the above mappings.

2. Representation of Multi-Radical Quadratic (Quartic) Mappings

In this section, we define multi-radical quadratic (quartic) mappings and present them as single equations.

Definition 1.

Let V be a linear vector space. A multivariable mapping $f:{\mathbb{R}}^n⟶V$ is said to be

•n-radical quadratic or multi-radical quadratic if it fulfils the general system of quadratic equations taken from Equation (3):

$$\begin{array}{cc}\hfill & f\left(u_1,\dots ,u_{i-1},\sqrt{u_i^2+v_i^2},u_{i+1},\dots ,u_n\right)\hfill \\ \hfill & =f(u_1,\dots ,u_{i-1},u_i,u_{i+1},\dots ,u_n)+f(u_1,\dots ,u_{i-1},v_i,u_{i+1},\dots ,u_n),\hfill \end{array}$$

•n-radical quadratic or multi-radical quadratic provided that it satisfies the general system of quartic equations taken from Equation (4):

$$\begin{array}{cc}\hfill f\left(u_1,\dots ,u_{i-1},\sqrt{u_i^2+v_i^2},u_{i+1},\dots ,u_n\right)& +f\left(u_1,\dots ,u_{i-1},\sqrt{|u_i^2-v_i^2|},u_{i+1},\dots ,u_n\right)\hfill \\ \hfill & =2f(u_1,\dots ,u_{i-1},u_i,u_{i+1},\dots ,u_n)\hfill \\ \hfill & +2f(u_1,\dots ,u_{i-1},v_i,u_{i+1},\dots ,u_n),\hfill \end{array}$$

for all $i\in \{1,\dots ,n\}$.

Clearly, the mapping $f:{\mathbb{R}}^n⟶\mathbb{R}$, defined via $f(r_1,\dots ,r_n)={\prod }_{j=1}^n{r}_j^2$ (resp. $f(r_1,\dots ,r_n)={\prod }_{j=1}^n{r}_j^4$), satisfies the system of n-radical quadratic (resp. n-radical quartic) functional equations indicated in Definition 1.

Here and subsequently, ${\mathbb{R}}_{+}=[0,\infty )$ denotes each element of ${\mathbb{R}}^n$ by $x^{\left[n\right]}$ and considers $x_i^{\left[n\right]}=(x_{i1},\dots ,x_{in})\in {\mathbb{R}}^n$, where $i\in \{1,2\}$. In the following result, we describe a multi-radical quadratic mapping as one equation.

Theorem 1.

A mapping $f:{\mathbb{R}}^n⟶V$ is multi-radical quadratic if and only if it fulfils

$$\begin{array}{c}\hfill f\left(\sqrt{x_{11}^2+x_{21}^2},\sqrt{x_{12}^2+x_{22}^2},\dots ,\sqrt{x_{1n}^2+x_{2n}^2}\right)=\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{j_{1}1},\dots ,x_{j_{n}n}),\end{array}$$

(6)

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$ described above.

Proof. 

Through a routine computation and by using induction (and maybe a direct extension), one can show that every multi-radical quadratic mapping fulfils (6). For other implications, we must first prove that $f\left(x^{\left[n\right]}\right)=0$ for any $x^{\left[n\right]}\in {\mathbb{R}}^n$ with at least one component which is equal to zero. By setting ${\mathbf{0}}_t=\stackrel{t-\mathrm{times}}{\overbrace{0,\dots ,0}}$ and making $x_1^{\left[n\right]}=x_2^{\left[n\right]}=\left({\mathbf{0}}_n\right)$ in (6), we obtain

$$\begin{array}{c}\hfill f\left({\mathbf{0}}_n\right)=2^{n}f\left({\mathbf{0}}_n\right),\end{array}$$

(7)

and thus $f\left({\mathbf{0}}_n\right)=0$. Letting $x_{1k}=0$ for all $k\in \{1,\dots ,n\}\setminus \left\{j\right\}$ and $x_{2k}=0$ for $1\le k\le n$ in (6) and using $f\left({\mathbf{0}}_n\right)=0$, we obtain

$$\begin{array}{c}\hfill f\left({\mathbf{0}}_{j-1},x_{1j},{\mathbf{0}}_{n-j}\right)=2^{n-1}f\left({\mathbf{0}}_{j-1},x_{1j},{\mathbf{0}}_{n-j}\right)\end{array}$$

Consequently, $f\left({\mathbf{0}}_{j-1},x_{1j},{\mathbf{0}}_{n-j}\right)=0$. We continue in this fashion, obtaining $f\left(x^{\left[n\right]}\right)=0$ for any $x^{\left[n\right]}\in {\mathbb{R}}^n$ with at least one component which is equal to zero. Lastly, setting $j\in \{1,\dots ,n\}$ and $x_{2k}=0$ for all $k\in \{1,\dots ,n\}\setminus \left\{j\right\}$ in (6), we have

$$\begin{array}{cc}\hfill f(x_{11},\dots ,x_{1,j-1},\sqrt{x_{1j}^2+x_{2j}^2},x_{1,j+1},\dots ,x_{1n})& =f(x_{11},\dots ,x_{1,j-1},x_{1j},x_{1,j+1},\dots ,x_{1n})\hfill \\ \hfill & +f(x_{11},\dots ,x_{1,j-1},x_{2j},x_{1,j+1},\dots ,x_{1n})\hfill \end{array}$$

which proves that f is a multi-radical quadratic mapping. □

Similar to the above, we present a characterization of the multi-radical quartic mappings below.

Theorem 2.

A mapping $f:{\mathbb{R}}^n⟶V$ is multi-radical quartic if and only it satisfies

$$\begin{array}{c}\hfill \sum _{{\mathfrak{V}}_j\in \{U_j,U_j^{\prime }\}}f\left({\mathfrak{V}}_1,\dots ,{\mathfrak{V}}_n\right)=2^n\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{j_{1}1},\dots ,x_{j_{n}n}),\end{array}$$

(8)

where

$$\begin{array}{c}\hfill U_j=\sqrt{x_{1j}^2+x_{2j}^2},U_j^{\prime }=\sqrt{|x_{1j}^2-x_{2j}^2|},\end{array}$$

(9)

for all $j\in \{1,\dots ,n\}$.

Proof. 

The proof of Theorem 1 can be repeated to achieve the desired results. □

3. Stability Results for Multi-Radical Quadratic (Quartic) Mappings

In this section, we prove the Găvruţa stability of multi-radical quadratic (quartic) mappings by using a fixed-point theorem in Banach spaces. In the following, for two sets, E and F, we denote the set of all mappings from E to F as $F^E$.

Theorem 3 

([25], Theorem 1). Let $n\in \mathbb{N}$. Suppose that $\mathbb{B}$ is a Banach space and $\mathcal{E}$ is a nonempty set. Given the functions $L_1,\dots ,L_n:\mathcal{E}⟶{\mathbb{R}}_{+}$ and the mappings $h_1,\dots ,h_n:\mathcal{E}⟶\mathcal{E}$, the following assumptions hold:

(H1) 

Consider $\mathcal{J}:{\mathbb{B}}^{\mathcal{E}}⟶{\mathbb{B}}^{\mathcal{E}}$ to be an operator such that

$$∥\mathcal{J}\lambda \left(e\right)-\mathcal{J}\mu \left(e\right)∥\le \sum _{i=1}^n{L}_i\left(e\right)∥\lambda \left(h_i\left(e\right)\right)-\mu \left(h_i\left(e\right)\right)∥,\lambda ,\mu \in {\mathbb{B}}^{\mathcal{E}},e\in \mathcal{E};$$

(H2) 

Let $\Lambda :{\mathbb{R}}_{+}^{\mathcal{E}}⟶{\mathbb{R}}_{+}^{\mathcal{E}}$ be an operator defined through

$$\Lambda \delta \left(e\right):=\sum _{i=1}^n{L}_i\left(e\right)\delta \left(h_i\left(e\right)\right)\delta \in {\mathbb{R}}_{+}^{\mathcal{E}},e\in \mathcal{E}.$$

Moreover, assume that a function $\theta :\mathcal{E}⟶{\mathbb{R}}_{+}$ and a mapping $\varphi :\mathcal{E}⟶\mathbb{B}$ fulfil the relations $\parallel \mathcal{J}\varphi \left(e\right)-\varphi \left(e\right)\parallel \le \theta \left(e\right)$ and ${\theta }^{*}\left(e\right):={\sum }_{l=0}^{\infty }{\Lambda }^l\theta \left(e\right)<\infty$ for all $e\in \mathcal{E}$. Then, there exists a unique fixed point ψ of $\mathcal{J}$ such that

$$\parallel \varphi \left(e\right)-\psi \left(e\right)\parallel \le {\theta }^{*}\left(e\right)(e\in \mathcal{E}).$$

Additionally, $\psi \left(e\right)={lim}_{l\to \infty }{\mathcal{J}}^l\varphi \left(e\right)$ for all $e\in \mathcal{E}$.

3.1. Stability Results for Multi-Radical Quadratic Mappings

Theorem 4.

Given $\alpha \in \{-1,1\}$ and V is a Banach space, we can take $\varphi :{\mathbb{R}}^n\times {\mathbb{R}}^n⟶{\mathbb{R}}_{+}$ as a function satisfying

$$\begin{array}{c}\hfill \underset{l\to \infty }{lim}{\left({\displaystyle \frac{1}{2^{n\alpha }}}\right)}^l\varphi \left({\sqrt{2}}^{l\alpha -{\displaystyle \frac{|\alpha -1|}{2}}}{x}_1^{\left[n\right]},{\sqrt{2}}^{l\alpha -{\displaystyle \frac{|\alpha -1|}{2}}}{x}_2^{\left[n\right]}\right)=0\end{array}$$

(10)

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$ and

$$\begin{array}{c}\hfill \Phi \left(x^{\left[n\right]}\right)=:{\displaystyle \frac{1}{2^{\frac{\alpha +1}{2}n}}}\sum _{l=0}^{\infty }{\left({\displaystyle \frac{1}{2^{n\alpha }}}\right)}^l\varphi \left({\sqrt{2}}^{l\alpha -{\displaystyle \frac{|\alpha -1|}{2}}}{x}^{\left[n\right]},{\sqrt{2}}^{l\alpha -{\displaystyle \frac{|\alpha -1|}{2}}}{x}^{\left[n\right]}\right)<\infty \end{array}$$

(11)

for all $x^{\left[n\right]}\in {\mathbb{R}}^n$. We also assume that $f:{\mathbb{R}}^n⟶V$ is a mapping fulfilling the inequality

$$\begin{gathered} ∥f\left(\sqrt{x_{11}^2+x_{21}^2},\sqrt{x_{12}^2+x_{22}^2},\dots ,\sqrt{x_{1n}^2+x_{2n}^2}\right)-\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{j_{1}1},\dots ,x_{j_{n}n})∥ \\ \le \varphi \left(x_1^{\left[n\right]},x_2^{\left[n\right]}\right) \end{gathered}$$

(12)

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$. Then, there exists a unique multi-radical mapping $\mathcal{R}:{\mathbb{R}}^n⟶V$ such that

$$\parallel f\left(x^{\left[n\right]}\right)-\mathcal{R}\left(x^{\left[n\right]}\right)\parallel \le \Phi \left(x^{\left[n\right]}\right)$$

(13)

for all $x^{\left[n\right]}\in {\mathbb{R}}^n$.

Proof. 

For $x^{\left[n\right]}\in {\mathbb{R}}^n$, we set $\theta \left(x^{\left[n\right]}\right):={\displaystyle \frac{1}{2^{\frac{\alpha +1}{2}n}}}\varphi \left({\displaystyle \frac{x^{\left[n\right]}}{{\sqrt{2}}^{\frac{|\alpha -1|}{2}}}},{\displaystyle \frac{x^{\left[n\right]}}{{\sqrt{2}}^{\frac{|\alpha -1|}{2}}}}\right)$ and $\mathcal{J}\theta \left(x^{\left[n\right]}\right):={\displaystyle \frac{1}{2^{n\alpha }}}\theta \left({\sqrt{2}}^{\alpha }{x}^{\left[n\right]}\right)$. Putting $x_1^{\left[n\right]}=x_2^{\left[n\right]}:=x^{\left[n\right]}$ into (12), we have

$$\begin{array}{c}\hfill ∥f\left(x^{\left[n\right]}\right)-\mathcal{J}f\left(x^{\left[n\right]}\right)∥\le \theta \left(x^{\left[n\right]}\right)\end{array}$$

(14)

for all $x^{\left[n\right]}\in {\mathbb{R}}^n$. We define $\Lambda \eta \left(x^{\left[n\right]}\right):={\displaystyle \frac{1}{2^{n\alpha }}}\eta \left({\sqrt{2}}^{\alpha }{x}^{\left[n\right]}\right)$ for all $\eta \in {\mathbb{R}}_{+}^{{\mathbb{R}}^n},x^{\left[n\right]}\in {\mathbb{R}}^n$. We now see that $\Lambda$ has the form shown in (H2). Moreover, for each $\lambda ,\mu \in V^{{\mathbb{R}}^n}$ and $x^{\left[n\right]}\in {\mathbb{R}}^n$, we obtain

$$\begin{array}{c}\hfill ∥\mathcal{J}\lambda \left(x^{\left[n\right]}\right)-\mathcal{J}\mu \left(x^{\left[n\right]}\right)∥\le {\displaystyle \frac{1}{2^{n\alpha }}}∥\eta \left({\sqrt{2}}^{\alpha }{x}^{\left[n\right]}\right)-\mu \left({\sqrt{2}}^{\alpha }{x}^{\left[n\right]}\right)∥.\end{array}$$

The above relation shows that the hypothesis (H1) of Theorem 3 holds. By inducting l, one can check for any $l\in {\mathbb{N}}_0=\mathbb{N}\bigcup \left\{0\right\}$ and $x^{\left[n\right]}\in {\mathbb{R}}^n$ that

$$\begin{array}{c}\hfill {\Lambda }^l\theta \left(x^{\left[n\right]}\right):={\left({\displaystyle \frac{1}{2^{n\alpha }}}\right)}^l\theta \left({\sqrt{2}}^{l\alpha }{x}^{\left[n\right]}\right).\end{array}$$

(15)

Now, Relations (11) and (15) necessitate that all assumptions of Theorem 3 are satisfied. Hence, there exists a mapping $\mathcal{R}:{\mathbb{R}}^n⟶V$ such that

$$\mathcal{R}\left(x^{\left[n\right]}\right)=\underset{l\to \infty }{lim}\left({\mathcal{J}}^{l}f\right)\left(x^{\left[n\right]}\right)={\displaystyle \frac{1}{2^n}}\mathcal{R}\left(\sqrt{2}{x}^{\left[n\right]}\right)\left(x^{\left[n\right]}\in {\mathbb{R}}^n\right),$$

and (13) also holds. Furthermore, we have

$$\begin{array}{c}∥{\mathcal{J}}^{l}f\left(\sqrt{x_{11}^2+x_{21}^2},\sqrt{x_{12}^2+x_{22}^2},\dots ,\sqrt{x_{1n}^2+x_{2n}^2}\right)-\sum _{j_1,\dots ,j_n\in \{1,2\}}{\mathcal{J}}^{l}f(x_{j_{1}1},\dots ,x_{j_{n}n})∥\hfill \\ \le {\left({\displaystyle \frac{1}{2^{n\alpha }}}\right)}^l\varphi \left({\sqrt{2}}^{l\alpha -{\displaystyle \frac{|\alpha -1|}{2}}}{x}_1^{\left[n\right]},{\sqrt{2}}^{l\alpha -{\displaystyle \frac{|\alpha -1|}{2}}}{x}_2^{\left[n\right]}\right)\hfill \end{array}$$

(16)

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$ and $l\in {\mathbb{N}}_0$. By letting $l\to \infty$ in (16) and applying (10), we arrive at

$$\mathcal{R}\left(\sqrt{x_{11}^2+x_{21}^2},\sqrt{x_{12}^2+x_{22}^2},\dots ,\sqrt{x_{1n}^2+x_{2n}^2}\right)=\sum _{j_1,\dots ,j_n\in \{1,2\}}\mathcal{R}(x_{j_{1}1},\dots ,x_{j_{n}n}),$$

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$. Now, Theorem 1 implies that $\mathcal{R}$ is a multi-radical mapping. Lastly, by assuming that $\mathfrak{R}:{\mathbb{R}}^n⟶V$ is another multi-radical mapping satisfying Equation (6) and Inequality (13), and fixing $x^{\left[n\right]}\in {\mathbb{R}}^n$, $j\in \mathbb{N}$, we obtain

$$\begin{array}{cc}\hfill & \parallel \mathcal{R}\left(x^{\left[n\right]}\right)-\mathfrak{R}\left(x^{\left[n\right]}\right)\parallel \hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{2^{n\alpha }}}\right)}^j\left(∥\mathcal{R}\left({\sqrt{2}}^{j\alpha }{x}^{\left[n\right]}\right)-f\left({\sqrt{2}}^{j\alpha }{x}^{\left[n\right]}\right)∥+∥\mathfrak{R}\left({\sqrt{2}}^{j\alpha }{x}^{\left[n\right]}\right)-f\left({\sqrt{2}}^{j\alpha }{x}^{\left[n\right]}\right)∥\right)\hfill \\ \hfill & \le 2{\left({\displaystyle \frac{1}{2^{n\alpha }}}\right)}^j{\displaystyle \frac{1}{2^{\frac{\alpha +1}{2}n}}}\sum _{l=j}^{\infty }{\left({\displaystyle \frac{1}{2^{n\alpha }}}\right)}^l\varphi \left({\sqrt{2}}^{l\alpha -{\displaystyle \frac{|\alpha -1|}{2}}}{x}^{\left[n\right]},{\sqrt{2}}^{l\alpha -{\displaystyle \frac{|\alpha -1|}{2}}}{x}^{\left[n\right]}\right).\hfill \end{array}$$

Consequently, by letting $j\to \infty$ and using the fact that Series (11) is convergent for all $x\in {\mathbb{R}}^n$, we obtain $\mathcal{R}\left(x^{\left[n\right]}\right)=\mathfrak{R}\left(x^{\left[n\right]}\right)$ for all $x^{\left[n\right]}\in {\mathbb{R}}^n$. This completes the proof. □

Next, we investigate the Rassias stability of multi-radical quadratic mappings.

Corollary 1.

Given $\delta \in [0,\infty ),r\in \mathbb{R}$ when $r\ne 2n$. Let V be a Banach space. Suppose that $f:{\mathbb{R}}^n⟶V$ is a mapping fulfilling the inequality

$$\begin{array}{c}\hfill ∥f\left(\sqrt{x_{11}^2+x_{21}^2},\sqrt{x_{12}^2+x_{22}^2},\dots ,\sqrt{x_{1n}^2+x_{2n}^2}\right)-\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{j_{1}1},\dots ,x_{j_{n}n})∥\hfill \\ \hfill \le \delta \sum _{i=1}^2\sum _{j=1}^n{\left|x_{ij}\right|}^r\hfill \end{array}$$

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$. Then, there exists a unique multi-radical mapping $\mathcal{R}:{\mathbb{R}}^n⟶V$ such that

$$∥f\left(x^{\left[n\right]}\right)-\mathcal{R}\left(x^{\left[n\right]}\right)∥\le {\displaystyle \frac{2\delta }{\left|2^n-{\sqrt{2}}^r\right|}}\sum _{j=1}^n{\left|x_{1j}\right|}^r$$

for all $x^{\left[n\right]}:=x_1^{\left[n\right]}\in {\mathbb{R}}^n$.

Proof. 

The result follows from Theorem 4 when considering

$$\varphi \left(x_1^{\left[n\right]},x_2^{\left[n\right]}\right)=\delta \sum _{i=1}^2\sum _{j=1}^n{\left|x_{ij}\right|}^r.$$

□

3.2. Stability Results for Multi-Radical Quartic Mappings

First, we highlight a fundamental result from [26].

Lemma 1.

Let $d\in \mathbb{N}$, $l\in {\mathbb{N}}_0$, $S:={\{0,1\}}^d$ and $\Delta :S⟶\mathbb{R}$. Suppose that

$$S_i:=\{(s_1,\dots ,s_d)\in S:Card\{j:s_j=0\}=i\},$$

for all $i\in \{0,\dots ,d\}$. Then,

$$\sum _{\alpha =0}^d\sum _{\beta =0}^d\sum _{s\in S_{\beta }}\sum _{t\in S_{\alpha }}{\left(2^l-1\right)}^{\beta }\Delta \left(st\right)=\sum _{i=0}^d\sum _{p\in S_i}{\left(2^{l+1}-1\right)}^i\Delta \left(p\right).$$

Lemma 1 can be used when the incoming stability result for multi-radical quartic mappings is associated with (4). For a mapping $f:{\mathbb{R}}^n⟶V$, we consider

$$\begin{array}{cc}\hfill \mathcal{D}f\left(x_1^{\left[n\right]},x_2^{\left[n\right]}\right):& =\sum _{{\mathfrak{V}}_j\in \{U_j,U_j^{\prime }\}}f\left({\mathfrak{V}}_1,\dots ,{\mathfrak{V}}_n\right)-2^n\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{j_{1}1},\dots ,x_{j_{n}n})\hfill \end{array}$$

where $U_j$ and $U_j^{\prime }$ are defined in (9).

Theorem 5.

Let $\psi :{\mathbb{R}}^n\times {\mathbb{R}}^n⟶{\mathbb{R}}_{+}$ be a mapping satisfying the equality

$$\begin{array}{c}\hfill \underset{l\to \infty }{lim}{\left({\displaystyle \frac{1}{2^{2n}}}\right)}^l\sum _{i=0}^n\sum _{p\in S_i}{\left(2^l-1\right)}^i\psi \left({\sqrt{2}}^{l}p{x}_1^{\left[n\right]},{\sqrt{2}}^{l}p{x}_2^{\left[n\right]}\right)=0\end{array}$$

(17)

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$ and

$$\begin{array}{c}\hfill \Psi \left(x^{\left[n\right]}\right)=:{\displaystyle \frac{1}{2^{2n}}}\sum _{l=0}^n{\left({\displaystyle \frac{1}{2^{2n}}}\right)}^l\sum _{i=0}^n\sum _{p\in S_i}{\left(2^l-1\right)}^i\psi \left({\sqrt{2}}^{l}p{x}^{\left[n\right]},{\sqrt{2}}^{l}p{x}^{\left[n\right]}\right)<\infty \end{array}$$

(18)

for all $x^{\left[n\right]}\in {\mathbb{R}}^n$. Assume also that the mapping $f:{\mathbb{R}}_{+}^n⟶V$ satisfies

$$\begin{array}{c}\hfill ∥\mathcal{D}f\left(x_1^{\left[n\right]},x_2^{\left[n\right]}\right)∥\le \psi \left(x_1^{\left[n\right]},x_2^{\left[n\right]}\right)\end{array}$$

(19)

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$, where $U_j$ and $U_j^{\prime }$ are defined in (9). Then, there exists a unique multi-radical quartic $\mathcal{Q}:{\mathbb{R}}^n⟶V$ such that

$$∥f\left(x^{\left[n\right]}\right)-\mathcal{Q}\left(x^{\left[n\right]}\right)∥\le \Psi \left(x^{\left[n\right]}\right),$$

(20)

for all $x^{\left[n\right]}\in {\mathbb{R}}^n$.

Proof. 

Similar to the proof of the previous theorem, by inserting $x_1^{\left[n\right]}=x_2^{\left[n\right]}:=x^{\left[n\right]}$ into (19), we have

$$\begin{array}{c}\hfill ∥f\left(x^{\left[n\right]}\right)-\mathcal{J}f\left(x^{\left[n\right]}\right)∥\le \xi \left(x^{\left[n\right]}\right),\end{array}$$

(21)

for all $x^{\left[n\right]}\in {\mathbb{R}}^n$, where

$$\xi \left(x^{\left[n\right]}\right):={\displaystyle \frac{1}{2^{2n}}}\psi \left(x^{\left[n\right]},x^{\left[n\right]}\right),\mathcal{J}\xi \left(x^{\left[n\right]}\right):={\displaystyle \frac{1}{2^{2n}}}\sum _{s\in S}\xi \left(\sqrt{2}s{x}^{\left[n\right]}\right),$$

in which $\xi \in V^{{\mathbb{R}}^n}$ and $x^{\left[n\right]}\in {\mathbb{R}}^n$, for which $S:={\{0,1\}}^n$ was taken from Lemma 1. By defining $\Lambda \eta \left(x^{\left[n\right]}\right):={\displaystyle \frac{1}{2^{2n}}}{\sum }_{s\in S}\eta \left(\sqrt{2}s{x}^{\left[n\right]}\right)$ for all $\eta \in V^{{\mathbb{R}}^n},x^{\left[n\right]}\in {\mathbb{R}}^n$, we observe that $\Lambda$ has the form shown in (H2). Additionally, for each $\lambda ,\mu \in V^{{\mathbb{R}}^n}$ and $x^{\left[n\right]}\in {\mathbb{R}}^n$, we find that

$$\begin{array}{c}\hfill ∥\mathcal{J}\lambda \left(x^{\left[n\right]}\right)-\mathcal{J}\mu \left(x^{\left[n\right]}\right)∥\le {\displaystyle \frac{1}{2^{2n}}}\sum _{s\in S}∥\lambda \left(\sqrt{2}s{x}^{\left[n\right]}\right)-\mu \left(\sqrt{2}s{x}^{\left[n\right]}\right)∥.\end{array}$$

This last relation implies that the assumption (H1) of Theorem 3 holds. Here, we claim that

$$\begin{array}{c}\hfill {\Lambda }^l\xi \left(x^{\left[n\right]}\right):={\left({\displaystyle \frac{1}{2^{2n}}}\right)}^l\sum _{i=0}^n{\left(2^l-1\right)}^i\sum _{p\in S_i}\xi \left({\sqrt{2}}^{l}p{x}^{\left[n\right]}\right),\end{array}$$

(22)

for all $l\in {\mathbb{N}}_0$ and $x^{\left[n\right]}\in {\mathbb{R}}^n$. The routine way to prove this is by means of an induction on l. We fix an $x^{\left[n\right]}\in {\mathbb{R}}^n$. To simplify, the convention that $0^0=1$ is adopted. Hence, (22) is true for $l=0$. We assume now that (22) is valid for a $l\in {\mathbb{N}}_0$. By applying Lemma 1 in the case where $m=n$ and $\psi \left(s\right):=\xi \left({\sqrt{2}}^{l+1}s{x}^{\left[n\right]}\right)(s\in S)$, we obtain

$$\begin{array}{cc}\hfill {\Lambda }^{l+1}\xi \left(x^{\left[n\right]}\right)& =\Lambda \left({\Lambda }^l\xi \right)\left(x^{\left[n\right]}\right)={\displaystyle \frac{1}{2^{2n}}}\sum _{v=0}^n\sum _{t\in S_v}\left({\Lambda }^l\xi \right)\left(\sqrt{2}t{x}^{\left[n\right]}\right)\hfill \\ \hfill & ={\left({\displaystyle \frac{1}{2^{2n}}}\right)}^{l+1}\sum _{v=0}^n\sum _{t\in S_v}\sum _{w=0}^n{\left(2^l-1\right)}^w\sum _{s\in S_w}\xi \left({\sqrt{2}}^{l+1}st{x}^{\left[n\right]}\right)\hfill \\ \hfill & ={\left({\displaystyle \frac{1}{2^{2n}}}\right)}^{l+1}\sum _{v=0}^n\sum _{w=0}^n\sum _{s\in S_w}\sum _{t\in S_v}{\left(2^l-1\right)}^w\xi \left({\sqrt{2}}^{l+1}st{x}^{\left[n\right]}\right)\hfill \\ \hfill & ={\left({\displaystyle \frac{1}{2^{2n}}}\right)}^{l+1}\sum _{i=0}^n\sum _{p\in S_i}{\left({\sqrt{2}}^{l+1}-1\right)}^i\xi \left({\sqrt{2}}^{l+1}p{x}^{\left[n\right]}\right).\hfill \end{array}$$

Therefore, (22) is true for any $l\in {\mathbb{N}}_0$ and $x^{\left[n\right]}\in {\mathbb{R}}^n$. Now, relations (18) and (22) necessitate that all assumptions of Theorem 3 hold and there therefore exists a mapping $\mathcal{Q}:{\mathbb{R}}^n⟶V$ such that

$$\mathcal{Q}\left(x^{\left[n\right]}\right)=\underset{l\to \infty }{lim}\left({\mathcal{J}}^{l}f\right)\left(x^{\left[n\right]}\right)={\displaystyle \frac{1}{2^{2n}}}\sum _{s\in S}\mathcal{Q}\left(\sqrt{2}s{x}^{\left[n\right]}\right),$$

for all $x^{\left[n\right]}\in {\mathbb{R}}^n$. Moreover, inequality (20) is valid. To show that $\mathcal{Q}$ is a multi-radical quartic, we firstly prove that

$$\begin{array}{c}\hfill ∥\sum _{{\mathfrak{V}}_j\in \{U_j,U_j^{\prime }\}}\left({\mathcal{J}}^{l}f\right)\left({\mathfrak{V}}_1,\dots ,{\mathfrak{V}}_n\right)-2^n\sum _{j_1,\dots ,j_n\in \{1,2\}}\left({\mathcal{J}}^{l}f\right)(x_{j_{1}1},\dots ,x_{j_{n}n})∥\hfill \\ \le {\left({\displaystyle \frac{1}{2^{2n}}}\right)}^l\sum _{i=0}^k\sum _{p\in S_i}{\left(2^l-1\right)}^i\psi \left({\sqrt{2}}^{l}p{x}^{\left[n\right]},{\sqrt{2}}^{l}p{x}^{\left[n\right]}\right),\hfill \end{array}$$

(23)

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$ and $l\in {\mathbb{N}}_0$. We argue this claim via the induction on l. Clearly, Inequality (23) is true for $l=0$ due to (19). We assume that (23) holds for an $l\in {\mathbb{N}}_0$. It follows from Lemma 1 with $m:=n$ and $\Delta \left(s\right):=\psi \left({\sqrt{2}}^{l+1}s{x}_1^{\left[n\right]},{\sqrt{2}}^{l+1}s{x}_2^{\left[n\right]}\right)$ for $s\in S$ that

$$\begin{array}{cc} \hfill & ∥\mathcal{D}\left({\mathcal{J}}^{l+1}f\right)\left(x_1^{\left[n\right]},x_2^{\left[n\right]}\right)∥\hfill \\ \hfill & ={\displaystyle \frac{1}{2^{2n}}}∥\sum _{s\in S}\mathcal{D}\left({\mathcal{J}}^{l}f\right)\left(\sqrt{2}s{x}_1^{\left[n\right]},\sqrt{2}s{x}_2^{\left[n\right]}\right)∥\hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{2^{2n}}}\right)}^{l+1}\sum _{s\in S}\sum _{i=0}^n\sum _{t\in S_i}{\left(2^l-1\right)}^i\psi \left({\sqrt{2}}^{l+1}st{x}_1^{\left[n\right]},{\sqrt{2}}^{l+1}st{x}_2^{\left[n\right]}\right)\hfill \\ \hfill & ={\left({\displaystyle \frac{1}{2^{2n}}}\right)}^{l+1}\sum _{i=0}^n\sum _{p\in S_i}{\left(2^{l+1}-1\right)}^i\psi \left({\sqrt{2}}^{l+1}p{x}_1^{\left[n\right]},{\sqrt{2}}^{l+1}p{x}_2^{\left[n\right]}\right)\hfill \end{array}$$

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$. Taking $l\to \infty$ in (23) and using (17), $\mathcal{D}\mathcal{Q}\left(x_1^{\left[n\right]},x_2^{\left[n\right]}\right)=0$ can be derived for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$, and therefore the claim is proved. Lastly, let $\mathfrak{Q}:V^n⟶W$ be another multi-radical quartic mapping fulfilling (20). By fixing $x^{\left[n\right]}\in {\mathbb{R}}^n$, $j\in \mathbb{N}$, (18) implies that

$$\begin{array}{cc}\hfill & ∥\mathcal{Q}\left(x^{\left[n\right]}\right)-\mathfrak{Q}\left(x^{\left[n\right]}\right)∥\hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{2^{2n}}}\right)}^j\left(∥\mathcal{Q}\left({\sqrt{2}}^j{x}^{\left[n\right]}\right)-f\left({\sqrt{2}}^j{x}^{\left[n\right]}\right)∥+∥\mathfrak{Q}\left({\sqrt{2}}^j{x}^{\left[n\right]}\right)-f\left({\sqrt{2}}^j{x}^{\left[n\right]}\right)∥\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{2^{2n-1}}}{\left({\displaystyle \frac{1}{2^{2n}}}\right)}^j\sum _{l=j}^{\infty }{\left({\displaystyle \frac{1}{2^{2n}}}\right)}^l\sum _{i=0}^n\sum _{p\in S_i}{(2^l-1)}^i\psi \left({\sqrt{2}}^{l}p{x}^{\left[n\right]},{\sqrt{2}}^{l}p{x}^{\left[n\right]}\right).\hfill \end{array}$$

Consequently, by letting $j\to \infty$, we obtain $\mathcal{Q}\left(x^{\left[n\right]}\right)=\mathfrak{Q}\left(x^{\left[n\right]}\right)$ for all $x^{\left[n\right]}\in {\mathbb{R}}^n$. □

The next corollary is a direct consequence of Theorem 4 for the Hyers stability of multi-radical quartic mappings.

Corollary 2.

Let $\epsilon >0$. Let the mapping $f:{\mathbb{R}}^n⟶V$ also fulfil the inequality

$$\begin{array}{c}\hfill ∥\sum _{{\mathfrak{V}}_j\in \{U_j,U_j^{\prime }\}}f\left({\mathfrak{V}}_1,\dots ,{\mathfrak{V}}_n\right)-2^n\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{j_{1}1},\dots ,x_{j_{n}n})∥\le \epsilon \end{array}$$

for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$, where $U_j$ and $U_j^{\prime }$ are defined in (9). Then, there exists a multi-radical quartic $\mathcal{Q}:{\mathbb{R}}^n⟶V$ such that

$$∥f\left(x^{\left[n\right]}\right)-\mathcal{Q}\left(x^{\left[n\right]}\right)∥\le {\displaystyle \frac{\epsilon }{2^n(2^n-1)}}$$

for all $x^{\left[n\right]}\in {\mathbb{R}}^n$.

Proof. 

Considering $\psi \left(x_1^{\left[n\right]},x_2^{\left[n\right]}\right)=\epsilon$ for all $x_1^{\left[n\right]},x_2^{\left[n\right]}\in {\mathbb{R}}^n$, and using Theorem 4, we have

$$\begin{array}{cc}\hfill \Psi \left(x^{\left[n\right]}\right)& ={\displaystyle \frac{1}{2^{2n}}}\sum _{l=0}^{\infty }{\left({\displaystyle \frac{1}{2^{2n}}}\right)}^l\sum _{i=0}^n\sum _{p\in S_i}{\left(2^l-1\right)}^i\psi \left({\sqrt{2}}^{l}p{x}^{\left[n\right]},{\sqrt{2}}^{l}p{x}^{\left[n\right]}\right)\hfill \\ \hfill & ={\displaystyle \frac{\epsilon }{2^{2n}}}\sum _{l=0}^{\infty }{\left({\displaystyle \frac{1}{2^{2n}}}\right)}^l\sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right){\left(2^l-1\right)}^i\times 1^{n-i}\hfill \\ \hfill & ={\displaystyle \frac{\epsilon }{2^n(2^n-1)}}.\hfill \end{array}$$

Therefore, we have reached the proof. □

4. Conclusions and Future Works

We have introduced multi-radical quadratic and multi-radical quartic mappings as two systems of radical functional equations and then expressed them as two equations. The benefit of presenting such systems as a single equation is that it allows us to prove the stability of multi-radical quadratic and multi-radical quartic mappings by means of only one functional inequality, and not a number of functional equations. Moreover, we established the Găvruţa stability of the multi-radical quadratic and multi-radical quartic mappings by using a fixed-point technique. In fact, we have extended some results from former works like [11], which are considered functional equations whose solutions are one-variable mappings. It is known that there are various versions of radical functional equations available in the literature. In future work, by using these equations, one could generalize a one-dimensional radical functional equation to multiple variables and study its stability in miscellaneous spaces. Are there some applications for the materials in the present paper in other fields, such as science and engineering?