On an equation characterizing multi-cubic mappings and its stability and hyperstability

TL;DR

This paper investigates multi-cubic mappings, establishes a functional equation characterizing them, and applies fixed point methods to prove their Hyers-Ulam stability and hyperstability.

Contribution

It introduces a functional equation for multi-cubic mappings and extends stability results, including hyperstability, using fixed point techniques.

Findings

Multi-cubic mappings satisfy a specific functional equation.

Hyers-Ulam stability is established for these mappings.

Multi-cubic functional equations can be hyperstable.

Abstract

In this paper, we introduce $n$-variables mappings which are cubic in each variable. We show that such mappings satisfy a functional equation. The main purpose is to extend the applications of a fixed point method to establish the Hyers-Ulam stability for the multi-cubic mappings. As a consequence, we prove that a multi-cubic functional equation can be hyperstable.

Equations

$$\sum _{t\in \{-1,1{\}}^n}f(x_1+t{x}_2)=2^n\sum _{j_1,j_2,\cdots ,j_n\in \{1,2\}}f(x_{1j_1},x_{2j_2},\cdots ,x_{n{j}_n})$$

$$C(x+2y)-3C(x+y)+3C(x)-C(x-y)=6C(y)$$

$$\mathfrak{C}(2x+y)+\mathfrak{C}(2x-y)=2\mathfrak{C}(x+y)+2\mathfrak{C}(x-y)+12\mathfrak{C}(x)$$

$${\mathcal{M}}_T^n:=\left\{{\mathfrak{N}}_n=(N_1,N_2,\cdots ,N_n)\in {\mathcal{M}}^n|\text{Card}\{N_j:N_j=x_{1j}\}=T\right\}.$$

$$f\left({\mathcal{M}}_T^n\right):=\sum _{{\mathfrak{N}}_n\in {\mathcal{M}}_T^n}f({\mathfrak{N}}_n),$$

$$f\left({\mathcal{M}}_T^n,z\right):=\sum _{{\mathfrak{N}}_n\in {\mathcal{M}}_T^n}f({\mathfrak{N}}_n,z)(z\in V).$$

$$h(2x)=8h(x).$$

$$\sum _{q\in \{-1,1{\}}^n}f(2x_1+q{x}_2)=\sum _{k=0}^n{2}^{n-k}12^{k}f({\mathcal{M}}_k^n),$$

$$\sum _{q\in \{-1,1{\}}^{n+1}}f(2x_1^{n+1}+q{x}_2^{n+1})$$

$$+2\sum _{q\in \{-1,1{\}}^n}f(2x_1^n+q{x}_2^n,x_{1n+1}-x_{2n+1})$$

$$+12\sum _{q\in \{-1,1{\}}^n}f(2x_1^n+q{x}_2^n,x_{1n+1})$$

$$=2\sum _{k=0}^n\sum _{q\in \{-1,1\}}{2}^{n-k}12^{k}f({\mathcal{M}}_k^n,x_{1n+1}+q{x}_{2n+1})$$

$$+12\sum _{k=0}^n{2}^{n-k}12^{k}f({\mathcal{M}}_k^n,x_{1n+1})$$

$$=\sum _{k=0}^{n+1}{2}^{n+1-k}12^{k}f({\mathcal{M}}_k^{n+1}).$$

$$f(z_1,\cdots ,z_{j-1},2z_j,z_{j+1},\cdots ,z_n)=2^{r}f(z_1,\cdots ,z_{j-1},z_j,z_{j+1},\cdots ,z_n),$$

$$2^{n-1}\times 2^{3(n-1)}[f\left(x_{11},\cdots ,x_{1j-1},2x_{1j}+x_{2j},x_{1j+1},\cdots ,x_{1n}\right)$$

$$+f\left(x_{11},\cdots ,x_{1j-1},2x_{1j}-x_{2j},x_{1j+1},\cdots ,x_{1n}\right)]$$

$$=2^{n-1}[f\left(2x_{11},\cdots ,2x_{1j-1},2x_{1j}+x_{2j},2x_{1j+1},\cdots ,2x_{1n}\right)$$

$$+f\left(2x_{11},\cdots ,2x_{1j-1},2x_{1j}-x_{2j},2x_{1j+1},\cdots ,2x_{1n}\right)].$$

$$f^{\ast }(x_{1j},x_{2j}):$$

$$+f\left(x_{11},\cdots ,x_{1j-1},x_{1j}-x_{2j},x_{1j+1},\cdots ,x_{1n}\right).$$

$$2^{n-1}\times 2^{3(n-1)}[f\left(x_{11},\cdots ,x_{1j-1},2x_{1j}+x_{2j},x_{1j+1},\cdots ,x_{1n}\right)$$

$$+f\left(x_{11},\cdots ,x_{1j-1},2x_{1j}-x_{2j},x_{1j+1},\cdots ,x_{1n}\right)]$$

$$=2^{n-1}\times 2^n{f}^{\ast }(x_{1j},x_{2j})$$

$$\displaystyle+\sum_{k=1}^{n-1}\left[\left(\begin{array}[]{ccccc}n-1\\ k-1\\ \end{array}\right)2^{2(n-k)}\times 12^{k}\right]f\left(x_{11},\cdots,x_{1n}\right)$$

$$\displaystyle+\sum_{k=1}^{n-1}\left[\left(\begin{array}[]{ccccc}n-1\\ k\\ \end{array}\right)2^{2(n-k)-1}\times 12^{k}\right]f^{*}(x_{1j},x_{2j})$$

$$+12^{n}f\left(x_{11},\cdots ,x_{1n}\right)$$

$$\displaystyle=\left[2^{2n-1}+\sum_{k=1}^{n-1}\left(\begin{array}[]{ccccc}n-1\\ k\\ \end{array}\right)2^{2(n-k)-1}\times 12^{k}\right]f^{*}(x_{1j},x_{2j})$$

$$\displaystyle+\left[12^{n}+\sum_{k=1}^{n-1}\left(\begin{array}[]{ccccc}n-1\\ k-1\\ \end{array}\right)2^{2(n-k)}\times 12^{k}\right]f\left(x_{11},\cdots,x_{1n}\right).$$

$$\displaystyle 2^{2n-1}+\sum_{k=1}^{n-1}\left(\begin{array}[]{ccccc}n-1\\ k\\ \end{array}\right)2^{2(n-k)-1}\times 12^{k}$$

$$=2^{2n-1}{\left(1+3\right)}^{n-1}=2^{4n-3}.$$

$$\displaystyle 12^{n}+\sum_{k=1}^{n-1}\left(\begin{array}[]{ccccc}n-1\\ k-1\\ \end{array}\right)2^{2(n-k)}\times 12^{k}$$

$$\displaystyle=12^{n}+3\times 2^{2n}\sum_{k=0}^{n-2}\left(\begin{array}[]{ccccc}n-1\\ k-1\\ \end{array}\right)3^{k}$$

$$\displaystyle=12^{n}+3\times 2^{2n}\left(\sum_{k=0}^{n-1}\left[\left(\begin{array}[]{ccccc}n-1\\ k-1\\ \end{array}\right)3^{k}\right]-3^{n-1}\right)$$

$$=12^n+3\times 2^{2n}\left((1+3{)}^{n-1}-3^{n-1}\right)$$

$$=12^n+3\times 2^{2n}\left(2^{2(n-1)}-3^{n-1}\right)$$

$$=12\times 2^{4(n-1)}.$$

$$f\left(x_{11},\cdots ,x_{1j-1},2x_{1j}+x_{2j},x_{1j+1},\cdots ,x_{1n}\right)$$

$$+f\left(x_{11},\cdots ,x_{1j-1},2x_{1j}-x_{2j},x_{1j+1},\cdots ,x_{1n}\right)$$

$$=2f^{\ast }(x_{1j},x_{2j})+12f\left(x_{11},\cdots ,x_{1n}\right)$$

$$\Vert \mathcal{T}\lambda (x)-\mathcal{T}\mu (x)\Vert \le \sum _{i=1}^j{L}_i(x)\Vert \lambda (g_i(x))-\mu (g_i(x))\Vert ,\lambda ,\mu \in Y^{\mathcal{S}},x\in \mathcal{S},$$

$$\Lambda \delta (x):=\sum _{i=1}^j{L}_i(x)\delta (g_i(x))\delta \in {\mathbb{R}}_{+}^{\mathcal{S}},x\in \mathcal{S}.$$

Full text

On an equation characterizing multi-cubic

mappings and its stability and hyperstability

Abasalt Bodaghi*∗* and Behrouz Shojaee*∗∗*

*∗*Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran

E-mail: abasalt.bodaghi@gmail.com

*∗∗*Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

E-mail: shoujaei@kiau.ac.ir

Abstract. In this paper, we introduce $n$-variables mappings which are cubic in each variable. We show that such mappings satisfy a functional equation. The main purpose is to extend the applications of a fixed point method to establish the Hyers-Ulam stability for the multi-cubic mappings. As a consequence, we prove that a multi-cubic functional equation can be hyperstable.

Key Words and Phrases: Banach space, Hyers-Ulam stability, multi-cubic mapping.

2010 Mathematics Subject Classification: 39B52, 39B82, 39B72.

1. Introduction

The study of stability problems for functional equations is related to a question of Ulam [18] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [12]. Later on, various generalizations and extension of Hyers’ result were ascertained by Aoki [1], Th. M. Rassias [17], J. M. Rassias [16] and Găvruţa [11] in different versions. Since then, the stability problems have been extensively investigated for a variety of functional equations and spaces.

Let $V$ be a commutative group, $W$ be a linear space, and $n\geq 2$ be an integer. Recall from [10] that a mapping $f:V^{n}\longrightarrow W$ is called multi-additive if it is additive (satisfies Cauchy’s functional equation $A(x+y)=A(x)+A(y)$) in each variable. Some facts on such mappings can be found in [15] and many other sources. In addition, $f$ is said to be multi-quadratic if it is quadratic (satisfies quadratic functional equation $Q(x+y)+Q(x-y)=2Q(x)+2Q(y)$) in each variable [9]. In [19], Zhao et al. proved that the mapping $f:V^{n}\longrightarrow W$ is multi-quadratic if and only if the following relation holds

[TABLE]

where $x_{j}=(x_{1j},x_{2j},\cdots,x_{nj})\in V^{n}$ with $j\in\{1,2\}$. In [10] and [9], Ciepliński studied the generalized Hyers-Ulam stability of multi-additive and multi-quadratic mappings in Banach spaces, respectively (see also [19]).

One of the functional equations in the field of stability of functional equations is the cubic functional equation

[TABLE]

which is introduced by J. M. Rassias in [16] for the first time. It is easy to see that the mapping $f(x)=ax^{3}$ satisfies (2). Thus, every solution of the cubic functional equation (2) is said to be a cubic mapping. Rassias established the Ulam-Hyers stability problem for these cubic mappings. The following alternative cubic functional equation

[TABLE]

has been introduced by Jun and Kim in [14]. They found out the general solution and proved the Hyers-Ulam stability for the functional equation (3); for other forms of the (generalized) cubic functional equations and their stabilities on the various Banach spaces refer to [3], [4], [5], [13].

In this paper, we define multi-cubic mappings and present a characterization of such mappings. In other words, we reduce the system of $n$ equations defining the multi-cubic mappings to obtain a single equation. We also prove the generalized Hyers-Ulam stability for multi-cubic functional equations by applying the fixed point method which was introduced and used for the first time by Brzdȩk et al., in [6]; for more applications of this approach for the satbility of multi-Cauchy-Jensen mappings in Banach spaces and 2-Banach spaces see [2] and [7], respectively.

2. Characterization of multi-cubic mappings

Throughout this paper, $\mathbb{N}$ stands for the set of all positive integers, $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\},\mathbb{R}_{+}:=[0,\infty),n\in\mathbb{N}$. For any $l\in\mathbb{N}_{0},m\in\mathbb{N}$, $t=(t_{1},\cdots,t_{m})\in\{-1,1\}^{m}$ and $x=(x_{1},\cdots,x_{m})\in V^{m}$ we write $lx:=(lx_{1},\cdots,lx_{m})$ and $tx:=(t_{1}x_{1},\cdots,t_{m}x_{m})$, where $ra$ stands, as usual, for the $r$th power of an element $a$ of the commutative group $V$.

From now on, let $V$ and $W$ be vector spaces over the rationals, $n\in\mathbb{N}$ and $x_{i}^{n}=(x_{i1},x_{i2},\cdots,x_{in})\in V^{n}$, where $i\in\{1,2\}$. We shall denote $x_{i}^{n}$ by $x_{i}$ if there is no risk of ambiguity. Let $x_{1},x_{2}\in V^{n}$ and $T\in\mathbb{N}_{0}$ with $0\leq T\leq n$. Put $\mathcal{M}^{n}=\left\{(N_{1},N_{2},\cdots,N_{n})|\,\,N_{j}\in\{x_{1j}\pm x_{2j},x_{1j}\}\right\}$, where $j\in\{1,\cdots,n\}$. Consider

[TABLE]

For $r\in\mathbb{R}$, we put $r\mathcal{M}_{T}^{n}=\left\{r\mathfrak{N}_{n}:\mathfrak{N}_{n}\in\mathcal{M}_{T}^{n}\right\}$. We say the mapping $f:V^{n}\longrightarrow W$ is $n$-multi-cubic or multi-cubic if $f$ is cubic in each variable (see the equation (3)). For such mappings, we use the following notations:

[TABLE]

[TABLE]

Remark 2.1. It is easily verified that if the mapping $h$ satisfies the equation (3), then

[TABLE]

But the converse is not true. Let $(\mathcal{A},\|\cdot\|)$ be a Banach algebra. Fix the vector $a_{0}$ in $\mathcal{A}$ (not necessarily unit). Define the mapping $h:\mathcal{A}\longrightarrow\mathcal{A}$ by $h(a)=\|a\|^{3}a_{0}$ for any $a\in\mathcal{A}$. Clearly, for each $x\in\mathcal{A}$, $h(2x)=8h(x)$ while the relation (3) does not hold for $h$ even if we put $x=0$ and $0\neq y$. Therefore, the condition (5) does not imply that $h$ is a cubic mapping.

Proposition 2.2. If the mapping $f:V^{n}\longrightarrow W$ is multi-cubic, then $f$ satisfies the equation

[TABLE]

*where $f\left(\mathcal{M}_{k}^{n}\right)$ is defined in (4). *

Proof.

We prove $f$ satisfies the equation (6) by induction on $n$. For $n=1$, it is trivial that $f$ satisfies the equation (3). If (6) is valid for some positive integer $n>1$, then,

[TABLE]

This means that (6) holds for $n+1$. ∎

In the sequel, \left(\begin{array}[]{ccccc}n\\ k\\ \end{array}\right) is the binomial coefficient defined for all $n,k\in\mathbb{N}_{0}$ with $n\geq k$ by $n!/(k!(n-k)!)$.

We say the mapping $f:V^{n}\longrightarrow W$ satisfies (has) the $r$-power condition in the $j$th variable if

[TABLE]

for all $(z_{1},\cdots,z_{n})\in V^{n}$. It follows from Remark 2.1 that the 3-power condition does not imply $f$ is cubic in the $j$th variable. Using this condition, we show that if $f$ satisfies the equation (6), then it is multi-cubic as follows:

Proposition 2.3. *If the mapping $f:V^{n}\longrightarrow W$ satisfies the equation (6) and $3$-power condition in each variable, then it is multi-cubic. *

Proof.

Fix $j\in\{1,\cdots,n\}$. Putting $x_{2k}=0$ for all $k\in\{1,\cdots,n\}\backslash\{j\}$ in the left side of (6) and using the assumption, we get

[TABLE]

Set

[TABLE]

By the above replacements in (6), it follows from (2) that

[TABLE]

On the other hand, we have

[TABLE]

In addition,

[TABLE]

The relations (2), (24) and (30) imply that

[TABLE]

This means that $f$ is cubic in the $j$th variable. Since $j$ is arbitrary, we obtain the desired result. ∎

3. Stability Results for (6)

In this section, we prove the generalized Hyers-Ulam stability of equation (6) by a fixed point result (Theorem 3.1) in Banach spaces. Throughout, for two sets $X$ and $Y$, the set of all mappings from $X$ to $Y$ is denoted by $Y^{X}$. We introduce the upcoming three hypotheses:

1. (A1)

$Y$ is a Banach space, $\mathcal{S}$ is a nonempty set, $j\in\mathbb{N}$, $g_{1},\cdots,g_{j}:\mathcal{S}\longrightarrow\mathcal{S}$ and $L_{1},\cdots,L_{j}:\mathcal{S}\longrightarrow\mathbb{R}_{+}$, 2. (A2)

$\mathcal{T}:Y^{\mathcal{S}}\longrightarrow Y^{\mathcal{S}}$ is an operator satisfying the inequality

[TABLE] 3. (A3)

$\Lambda:\mathbb{R}_{+}^{\mathcal{S}}\longrightarrow\mathbb{R}_{+}^{\mathcal{S}}$ is an operator defined through

[TABLE]

Here, we highlight the following theorem which is a fundamental result in fixed point theory [6, Theorem 1]. This result plays a key tool to obtain our objecive in this paper.

Theorem 3.1. Let hypotheses (A1)-(A3) hold and the function $\theta:\mathcal{S}\longrightarrow\mathbb{R}_{+}$ and the mapping $\phi:\mathcal{S}\longrightarrow Y$ fulfill the following two conditions:

[TABLE]

Then, there exists a unique fixed point $\psi$ of $\mathcal{T}$ such that

[TABLE]

Moreover, $\psi(x)=\lim_{l\rightarrow\infty}\mathcal{T}^{l}\phi(x)$ for all $x\in\mathcal{S}$.

Here and subsequently, for the mapping $f:V^{n}\longrightarrow W$, we consider the difference operator $\mathfrak{D}f:V^{n}\times V^{n}\longrightarrow W$ by

[TABLE]

where $f\left(\mathcal{M}_{k}^{n}\right)$ is defined in (4). With this notation, we have the next stability result for the functional equation (6).

Theorem 3.2. Let $\beta\in\{-1,1\}$, let $V$ be a linear space and $W$ be a Banach space. Suppose that $\phi:V^{n}\times V^{n}\longrightarrow\mathbb{R}_{+}$ is a mapping satisfying

[TABLE]

for all $x_{1},x_{2}\in V^{n}$ and

[TABLE]

for all $x\in V^{n}$. Assume also $f:V^{n}\longrightarrow W$ is a mapping satisfying the inequality

[TABLE]

for all $x_{1},x_{2}\in V^{n}$. Then, there exists a unique multi-cubic mapping $\mathcal{C}:V^{n}\longrightarrow W$ such that

[TABLE]

for all $x\in V^{n}$.

Proof.

Putting $x=x_{1}$ and $x_{2}=0$ in (40), we have

[TABLE]

for all $x\in V^{n}$. By an easy computation, we have

[TABLE]

It follows from (44) and (49) that

[TABLE]

for all $x\in V^{n}$. Set

[TABLE]

Then, the relation (50) can be modified as

[TABLE]

Define $\Lambda\eta(x):=\frac{1}{2^{3n\beta}}\eta(2^{\beta}x)$ for all $\eta\in\mathbb{R}_{+}^{V^{n}},x\in V^{n}$. We now see that $\Lambda$ has the form described in (A3) with $\mathcal{S}=V^{n}$, $g_{1}(x)=2^{\beta}x$ and $L_{1}(x)=\frac{1}{2^{3n\beta}}$ for all $x\in V^{n}$. Furthermore, for each $\lambda,\mu\in W^{V^{n}}$ and $x\in V^{n}$, we get

[TABLE]

The above relation shows that the hypotheis (A2) holds. By induction on $l$, one can check that for any $l\in\mathbb{N}_{0}$ and $x\in V^{n}$, we have

[TABLE]

for all $x\in V^{n}$. The relations (39) and (52) necessitate that all assumptions of Theorem 3.1 are satisfied. Hence, there exsits a unique mapping $\mathcal{C}:V^{n}\longrightarrow W$ such that

[TABLE]

and (41) holds. We shall to show that

[TABLE]

for all $x_{1},x_{2}\in V^{n}$ and $l\in\mathbb{N}_{0}$. We argue by induction on $l$. The inequality (53) is valid for $l=0$ by (40). Assume that (53) is true for an $l\in\mathbb{N}_{0}$. Then

[TABLE]

for all $x_{1},x_{2}\in V^{n}$. Letting $l\rightarrow\infty$ in (53) and applying (38), we arrive at $\mathfrak{D}\mathcal{C}(x_{1},x_{2})=0$ for all $x_{1},x_{2}\in V^{n}$. This means that the mapping $\mathcal{C}$ satisfies (6). Finally, assume that $\mathcal{C}^{\prime}:V^{n}\longrightarrow W$ is another multi-cubic mapping satisfying the equation (6) and inequality (41), and fix $x\in V^{n}$, $j\in\mathbb{N}$. Then

[TABLE]

Consequently, letting $j\rightarrow\infty$ and using the fact that series (39) is convergent for all $x\in V^{n}$, we obtain $\mathcal{C}(x)=\mathcal{C}^{\prime}(x)$ for all $x\in V^{n}$, which finishes the proof. ∎

Let $A$ be a nonempty set, $(X,d)$ a metric space, $\psi\in\mathbb{R}_{+}^{A^{n}}$, and $\mathcal{F}_{1},\mathcal{F}_{2}$ operators mapping a nonempty set $D\subset X^{A}$ into $X^{A^{n}}$. We say that operator equation

[TABLE]

is $\psi$-hyperstable provided every $\varphi_{0}\in D$ satisfying inequality

[TABLE]

fulfils (55); this definition is introduced in [8]. In other words, a functional equation $\mathcal{F}$ is hyperstable if any mapping $f$ satisfying the equation $\mathcal{F}$ approximately is a true solution of $\mathcal{F}$.

Under some conditions the functional equation (6) can be hyperstable as follows.

Corollary 3.4. Let $\delta>0$. Suppose that $p_{ij}>0$ for $i\in\{1,2\}$ and $j\in\{1,\cdots,n\}$ fulfill $\sum_{i=1}^{2}\sum_{j=1}^{n}p_{ij}\neq 3n$. Let $V$ be a normed space and let $W$ be a Banach space. If $f:V^{n}\longrightarrow W$ is a mapping satisfying the inequality

[TABLE]

for all $x_{1},x_{2}\in V^{n}$, then $f$ is multi-cubic.

In the following corollary, we show that the functional equation (6) is stable. Since the proof is routine, we include it without proof.

Corollary 3.5. Let $\delta>0$ and $\alpha\in\mathbb{R}$ with $\alpha\neq 3n$. Let also $V$ be a normed space and let $W$ be a Banach space. If $f:V^{n}\longrightarrow W$ is a mapping satisfying the inequality

[TABLE]

for all $x_{1},x_{2}\in V^{n}$, then there exists a unique multi-cubic mapping $\mathcal{C}:V^{n}\longrightarrow W$ such that

[TABLE]

*for all $x\in V^{n}$. *

Acknowledgements

The authors sincerely appreciate the anonymous reviewer for her/his careful reading, constructive comments and fruitful suggestions to improve the paper.