Hyers–Ulam stability and hyperstability of a Jensen-type functional equation on 2-Banach spaces

Abstract

The main aim of this paper is to establish the Hyers–Ulam stability and hyperstability of a Jensen-type quadratic mapping in 2-Banach spaces. That is, we prove the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form

$$ g \biggl( \frac{x+y}{2} + z \biggr) + g \biggl( \frac{x+y}{2} - z \biggr) + g \biggl( \frac{x-y}{2} + z \biggr) + g \biggl( \frac{x-y}{2} - z \biggr) = g(x) + g(y) + 4 g(z), $$

in 2-Banach spaces by using the Hyers direct method.

1 Introduction and preliminaries

The concept of stability in a functional equation is introduced by substituting the corresponding functional inequality, which acts as a perturbation of the functional equation. In 1940, S.M. Ulam [23] was the first to propose the stability of homomorphisms between groups. The first affirmative solution to this question was given by D.H. Hyers [10] in 1941. Hyers’ theorem for additive and linear mappings was generalized independently by T. Aoki [1] and Th.M. Rassias [19]. In 1994, Găvruta [9] provided a more general theorem than the previous results.

It is well known that the parallelogram low

$$ \Vert a+b \Vert ^{2} + \Vert a-b \Vert ^{2} = 2 \Vert a \Vert ^{2} + 2 \Vert b \Vert ^{2}$$

is valid for the square norm on an inner product space. The functional equation $f(x+y) + f(x-y) = 2 f(x) + 2 f(y)$, which is derived from the parallelogram low, is said to be the quadratic functional equation. A function g between two groups (or linear spaces) is called a quadratic function if g is a solution of the quadratic functional equation.

Skof [22] proved that the quadratic functional equation has the Hyers–Ulam stability for functions $f: X \to Y$, where X is a normed space and Y is a Banach space. It should be noted that the stability theorem of Skof is still valid if an Abelian group replaces the normed space X (see [3]). A generalized Hyers–Ulam stability of the quadratic functional equation was obtained later by Czerwik [4] in Banach spaces. Some quadratic type functional equations have been introduced by several authors, and they obtained many interesting results concerning the Hyers–Ulam stability (see for example [2, 6, 12, 14–16, 20, 21] and the references therein).

Park et al. [17] introduced the following Jensen-type quadratic functional equation:

$$ \begin{aligned} &g \biggl(\frac{x+y}{2}+z \biggr)+g \biggl(\frac{x+y}{2}-z \biggr)+g \biggl(\frac{x-y}{2}+z \biggr)+g \biggl(\frac{x-y}{2}-z \biggr) \\ &\quad =g(x)+g(y)+4g(z), \end{aligned} $$

(1.1)

where $g:X\to Y$ is a function between linear spaces X and Y. They showed that if an even function $g:X\to Y$ satisfies $g(0)=0$ and (1.1) for all $x,y,z\in X$, then g is quadratic. We prove, without any additional assumptions, that if a function g satisfies (1.1), then g is quadratic.

In this paper, we investigate the Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation (1.1) in 2-Banach spaces.

The concept of linear 2-normed spaces was introduced by S. Gähler [7, 8] in the middle of 1960s. In the following, we recall some basic facts about 2-normed spaces and some preliminary results.

Definition 1.1

([7])

Let A be a linear space over $\mathbb{R}$ with $\dim A>1$. A function $\Vert \cdot ,\cdot \Vert : A\times A\rightarrow \mathbb{R}$ is called a 2-norm on A if it satisfies the following properties:

• (1) $\Vert a,b \Vert =0$ if and only if a and b are linearly dependent,

• (2) $\Vert \lambda a,b \Vert = \vert \lambda \vert \Vert a,b \Vert$,

• (3) $\Vert a,b \Vert = \Vert b,a \Vert$,

• (4) $\Vert a,b+c \Vert \leqslant \Vert a,b \Vert + \Vert a,c \Vert$,

for each $a,b,c\in A$ and $\lambda \in \mathbb{R}$. In this case, $(A, \Vert \cdot ,\cdot \Vert )$ is a called a linear 2-normed space. By $(1)$, $(3)$, and $(4)$, we infer that $\Vert a,b \Vert \geqslant 0$ for all $a,b\in A$.

Definition 1.2

Let A be a linear 2-normed space and $\{a_{n}\}_{n}$ be a sequence in A.

• The sequence $\{a_{n}\}_{n}$ is called a Cauchy sequence if

  $$ \lim_{l,m \rightarrow \infty } \Vert a_{l}-a_{m},y \Vert =0,\quad y\in A.$$

• The sequence $\{a_{n}\}_{n}$ is called a convergent sequence if there is $a\in A$ such that

  $$ \lim_{n \rightarrow \infty } \Vert a_{n}-a,b \Vert =0,\quad b \in A.$$

  In this case, a is called the limit of $\{a_{n}\}$, and we write $\lim_{n \rightarrow \infty }a_{n}=a$.

• A is called a 2-Banach space if each Cauchy sequence in A is convergent.

We need the following lemma, and we will use it in our results.

Lemma 1.3

[18] Let $(A, \Vert \cdot ,\cdot \Vert )$ be a 2-normed space.

$(i)$:

If $\Vert a,x \Vert =0$ for all $x\in A$, then $a=0$.

$(\mathit{ii})$:

For a convergent sequence $\{a_{n}\}$ in A,

$$ \lim_{n\rightarrow \infty } \Vert a_{n},x \Vert = \Bigl\Vert \lim_{n\rightarrow \infty } a_{n},x \Bigr\Vert ,\quad x \in A.$$

2 General solution of (1.1)

In this section, we provide the general solution of functional equation (1.1) without any additional assumptions. We also provide a Jordan and von Neumann type characterization theorem for inner product spaces. Throughout this part, $(G,+)$ is a 2-divisible group (not necessarily Abelian) and $(H,+)$ is an Abelian group.

Theorem 2.1

Suppose that $g: G \to H$ satisfies (1.1) for all $x,y,z\in G$. Then g is quadratic.

Proof

Letting $x=y=z=0$ in (1.1), we get $2g(0)=0$. Putting $z=0$ in (1.1) and using $2g(0)=0$, we obtain

$$ 2g \biggl(\frac{x+y}{2} \biggr)+2g \biggl( \frac{x-y}{2} \biggr)=g(x)+g(y), \quad x,y\in G. $$

(2.1)

Letting $y=0$ in (2.1), we get

$$ 4g \biggl(\frac{x}{2} \biggr)=g(x)+g(0),\quad x\in G. $$

(2.2)

It follows from (2.1) and (2.2) that

$$ 2g(x)+2g(y)= 4g \biggl(\frac{x+y}{2} \biggr)+4g \biggl( \frac{x-y}{2} \biggr)=g(x+y)+g(x-y)+2g(0),\quad x,y\in G.$$

Since $2g(0)=0$, we infer that g is quadratic. □

By [13, Theorem 4.1] and Theorem 2.1, we get the general solution of (1.1).

Corollary 2.2

Let $(H,+)$ be a 2-divisible Abelian group. Suppose that $g: G \to H$ satisfies (1.1) and $g(x + y + z) = g(x + z + y)$ for all $x,y,z\in G$. Then g has the form $g(x)=B(x,x)$, where $B:G\times G\to H$ is a symmetric and biadditive function. This function B is unique and is given by

$$ B(x, y)=\frac{g(x+y)-g(x-y)}{4}=\frac{g(x+y)-g(x)-g(y)}{2},\quad x,y \in G.$$

Now we give a Jordan and von Neumann type characterization theorem for inner product spaces which is related to (1.1).

Theorem 2.3

Let X be a normed space and

$$ \begin{aligned} & \biggl\Vert \frac{x+y}{2}+z \biggr\Vert ^{p}+ \biggl\Vert \frac{x+y}{2}-z \biggr\Vert ^{q}+ \biggl\Vert \frac{x-y}{2}+z \biggr\Vert ^{r}+ \biggl\Vert \frac{x-y}{2}-z \biggr\Vert ^{s} \\ &\quad = \Vert x \Vert ^{\alpha }+ \Vert y \Vert ^{\beta }+4 \Vert z \Vert ^{\gamma },\quad x,y,z\in X, \end{aligned} $$

(2.3)

for some nonnegative real numbers p, q, r, s, α, β, γ. Then X is an inner product space.

Proof

Letting $z=0$ in (2.3), we get

$$ \biggl\Vert \frac{x+y}{2} \biggr\Vert ^{p}+ \biggl\Vert \frac{x+y}{2} \biggr\Vert ^{q}+ \biggl\Vert \frac{x-y}{2} \biggr\Vert ^{r}+ \biggl\Vert \frac{x-y}{2} \biggr\Vert ^{s} = \Vert x \Vert ^{\alpha }+ \Vert y \Vert ^{\beta },\quad x,y,z\in X. $$

(2.4)

Letting $y=0$ and $\Vert x \Vert =2$ ($x=0$ and $\Vert y \Vert =2$) in (2.4), we obtain $\alpha =\beta =2$. Replacing $y=x$ in (2.4) and then setting $\Vert x \Vert =2,4$, we get $2^{p}+2^{q}=4$ and $4^{p}+4^{q}=32$. Then it is easily obtained $p=q=2$. Similarly, replacing $y=-x$ in (2.4) and then setting $\Vert x \Vert =2,4$, we obtain $r=s=2$. Therefore (2.4) means

$$ \Vert x+y \Vert ^{2}+ \Vert x-y \Vert ^{2}=2 \Vert x \Vert ^{2}+2 \Vert y \Vert ^{2},\quad x,y \in X.$$

Hence X is an inner product space by the Jordan-von Neumann result [11]. □

Theorem 2.4

Let X be a normed space and

$$ \begin{gathered} \biggl\Vert \frac{x+y}{2}+z \biggr\Vert ^{p}+ \biggl\Vert \frac{x+y}{2}-z \biggr\Vert ^{q}+ \biggl\Vert \frac{x-y}{2}+z \biggr\Vert ^{r}+ \biggl\Vert \frac{x-y}{2}-z \biggr\Vert ^{s}=6,\\ \Vert x \Vert = \Vert y \Vert = \Vert z \Vert =1,\end{gathered} $$

(2.5)

for some nonnegative real numbers p, q, r, s. Then X is an inner product space.

Proof

Letting $y=x$ in (2.5), we get

$$ \Vert x+z \Vert ^{p}+ \Vert x-z \Vert ^{q}=4,\quad\quad \Vert x \Vert = \Vert z \Vert =1. $$

(2.6)

Replacing $z=x$ ($z=-x$) in (2.6), we get $p=q=2$. Then (2.6) means

$$ \Vert x+z \Vert ^{2}+ \Vert x-z \Vert ^{2}=4, \quad\quad \Vert x \Vert = \Vert z \Vert =1.$$

Hence X is an inner product space by [5, Theorem 2.1]. □

3 Hyperstability and stability of (1.1)

Throughout this section, $\mathcal{A}$ denotes a linear 2-normed space, and $(G,+)$ is a 2-divisible Abelian group. For convenience, we set

$$\begin{aligned} D_{g}(x,y, z):={}& g \biggl( \frac{x+y}{2} + z \biggr) + g \biggl( \frac{x+y}{2} - z \biggr) + g \biggl( \frac{x-y}{2} + z \biggr) \\ & {}+ g \biggl( \frac{x-y}{2} - z \biggr) - g(x) - g(y) - 4 g(z) \end{aligned}$$

for a given function g.

We introduce some hyperstability results for the Jensen-type quadratic functional equation (1.1).

Theorem 3.1

Let $\varphi :G\to \mathcal{A}$ be a surjective function. Assume that $\eta : G^{4}\to [0,+\infty )$ and $g : G \to \mathcal{A}$ are functions satisfying

$$\begin{aligned} & \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \eta (x,y,z,w), \end{aligned}$$

(3.1)

$$\begin{aligned} & \lim_{n\to \infty }\eta \bigl((n+2)x,ny,0,w \bigr)=0, \quad\quad \lim_{n\to \infty } \eta (nx,ny,0,w)=0 \end{aligned}$$

(3.2)

for all $x,y , z, w \in G$. Then g is quadratic and satisfies functional equation (1.1); that is, $D_{g} (x, y, z)=0$ for all $x,y,z\in G$.

Proof

Letting $y=nx$, $z=0$ and replacing x by $(n+2)x$ in (3.1), we obtain

$$ \bigl\Vert 2g \bigl((n+1)x \bigr)+2g(x)-g \bigl((n+2)x \bigr)-g(nx)-4g(0), \varphi (w) \bigr\Vert \leqslant \eta \bigl((n+2)x,nx,0,w \bigr) $$

for all $x, y, w \in G$ and $n\in \mathbb{N}$. Therefore

$$ 2g(x)=\lim_{n\to \infty } \bigl[g \bigl((n+2)x \bigr)+g(nx)-2g \bigl((n+1)x \bigr)+4g(0) \bigr],\quad x \in G.$$

By (3.1) and (3.2), we get

$$\begin{aligned} & \biggl\Vert 2g \biggl(\frac{x+y}{2} \biggr)+2g \biggl( \frac{x-y}{2} \biggr)-g(x)-g(y)-4g(0),\varphi (w) \biggr\Vert \\ &\quad \leqslant \frac{1}{2}\limsup_{n\to \infty } \bigl\Vert D_{g} \bigl((n+2)x,(n+2)y,0 \bigr), \varphi (w) \bigr\Vert \\ &\quad\quad {} + \frac{1}{2}\limsup_{n\to \infty } \bigl\Vert D_{g}(nx,ny,0),\varphi (w) \bigr\Vert \\ &\quad\quad {} + \limsup_{n\to \infty } \bigl\Vert D_{g} \bigl((n+1)x,(n+1)y,0 \bigr),\varphi (w) \bigr\Vert \\ &\quad \leqslant \frac{1}{2}\limsup_{n\to \infty }\eta \bigl((n+2)x,(n+2)y,0,w \bigr) \\ &\quad\quad {} + \frac{1}{2}\limsup_{n\to \infty }\eta (nx,ny,0,w)+\limsup_{n \to \infty }\eta \bigl((n+1)x,(n+1)y,0,w \bigr)=0. \end{aligned}$$

Then

$$ 2g \biggl(\frac{x+y}{2} \biggr)+2g \biggl( \frac{x-y}{2} \biggr)=g(x)+g(y)+4g(0), \quad x,y\in G. $$

(3.3)

Letting $x=y=z=0$ in (3.3), we get $2g(0)=0$. The rest of the proof follows from the proof of Theorem 2.1. □

Remark 3.2

By a similar argument, it can be shown that if the conditions

$$ \lim_{n\to \infty }\eta \bigl(2(n+1)x,0,nx,w \bigr)=0, \quad\quad \lim_{n\to \infty } \eta (nx,0,ny,w)=0,\quad x,y,w\in G$$

are substituted for (3.2), Theorem 3.1 is still valid.

Corollary 3.3

Let $p,q<0$ and $r,\alpha ,\beta ,\gamma \geqslant 0$ be real numbers. Suppose that X is a linear normed space, $\psi :X\to [0,+\infty )$ and $g,\varphi : X \to \mathcal{A}$ are functions such that φ is surjective and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \bigl(\alpha \Vert x \Vert ^{p}+\beta \Vert y \Vert ^{q}+\gamma \Vert z \Vert ^{r} \bigr)\psi (w) $$

for all $x,y \in X\setminus \{0\}$ and $z, w\in X$. Then $D_{g} (x, y, z)=0$ for all $x,y\in X\setminus \{0\}$ and $z\in X$. In particular,

$$ 2g \biggl(\frac{x+y}{2} \biggr)+2g \biggl(\frac{x-y}{2} \biggr)=g(x)+g(y), \quad x,y\in X\setminus \{0\}.$$

Corollary 3.4

Let $p,r<0$ and $q,\alpha ,\beta ,\gamma \geqslant 0$ be real numbers. Suppose that X is a linear normed space, $\psi :X\to [0,+\infty )$ and $g,\varphi : X \to \mathcal{A}$ are functions such that φ is surjective and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \bigl(\alpha \Vert x \Vert ^{p}+\beta \Vert y \Vert ^{q}+\gamma \Vert z \Vert ^{r} \bigr)\psi (w) $$

for all $x,z \in X\setminus \{0\}$ and $y, w\in X$. Then $D_{g} (x, y, z)=0$ for all $x,z\in X\setminus \{0\}$ and $y\in X$. In particular,

$$ 2g \biggl(\frac{x+z}{2} \biggr)+2g \biggl(\frac{x-z}{2} \biggr)=g(x)+g(z)+2g(0), \quad x,z\in X\setminus \{0\}.$$

Theorem 3.5

Let $\varphi :G\to \mathcal{A}$ be a surjective function. Assume that $\eta : G^{4}\to [0,+\infty )$ and $g : G \to \mathcal{A}$ are functions satisfying

$$\begin{aligned} & \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \eta (x,y,z,w), \end{aligned}$$

(3.4)

$$\begin{aligned} & \lim_{n\to \infty }\eta \bigl((n+1)x,(n+1)x,nx,w \bigr)=0, \quad \quad \lim_{n\to \infty }\eta (nx,ny,nz,w)=0 \end{aligned}$$

(3.5)

for all $x,y , z, w \in G$. Then g is quadratic and satisfies functional equation (1.1); that is, $D_{g} (x, y, z)=0$ for all $x,y,z\in G$.

Proof

Letting $y=(n+1)x$, $z=nx$ and replacing x by $(n+1)x$ in (3.4), we obtain

$$ \bigl\Vert g \bigl((2n+1)x \bigr)+g(x)+g(-nx)-2g \bigl((n+1)x \bigr)-3g(nx), \varphi (w) \bigr\Vert \leqslant \eta \bigl((n+1)x,(n+1)x,nx,w \bigr) $$

for all $x, y, w \in G$ and $n\in \mathbb{N}$. Therefore

$$ g(x)=\lim_{n\to \infty } \bigl[2g \bigl((n+1)x \bigr)+3g(nx)-g \bigl((2n+1)x \bigr)-g(-nx) \bigr],\quad x \in G.$$

Hence (3.4) and (3.5) yield

$$\begin{aligned} \bigl\Vert D_{g} (x, y, z),\varphi (w) \bigr\Vert &\leqslant 2 \limsup_{n\to \infty } \bigl\Vert D_{g} \bigl((n+1)x,(n+1)y,(n+1)z \bigr),\varphi (w) \bigr\Vert \\ &\quad {} + 3\limsup_{n\to \infty } \bigl\Vert D_{g}(nx,ny,nz), \varphi (w) \bigr\Vert \\ &\quad {} + \limsup_{n\to \infty } \bigl\Vert D_{g} \bigl((2n+1)x,(2n+1)y,(2n+1)z \bigr), \varphi (w) \bigr\Vert \\ &\quad {} + \limsup_{n\to \infty } \bigl\Vert D_{g}(-nx,-ny,-nz), \varphi (w) \bigr\Vert \\ &\leqslant 2\limsup_{n\to \infty }\eta \bigl((n+1)x,(n+1)y,(n+1)z,w \bigr) \\ &\quad {} + 3\limsup_{n\to \infty }\eta (nx,ny,nz,w) \\ &\quad {} +\limsup_{n\to \infty }\eta \bigl((2n+1)x,(2n+1)y,(2n+1)z,w \bigr) \\ &\quad {} +\limsup_{n\to \infty }\eta (-nx,-ny,-nz,w)=0. \end{aligned}$$

Then $D_{g} (x, y, z)=0$ for all $x,y,z\in G$, which yields that g is quadratic by Theorem 2.1. □

Corollary 3.6

Let $p,q,r<0$ and $\alpha ,\beta ,\gamma \geqslant 0$ be real numbers. Suppose that X is a linear normed space, $\psi :X\to [0,+\infty )$ and $g,\varphi : X \to \mathcal{A}$ are functions such that φ is surjective and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \bigl(\alpha \Vert x \Vert ^{p}+\beta \Vert y \Vert ^{q}+\gamma \Vert z \Vert ^{r} \bigr)\psi (w) $$

for all $x,y , z\in X\setminus \{0\}$ and $w\in X$. Then $D_{g} (x, y, z)=0$ for all $x,y,z\in X\setminus \{0\}$.

Let us consider A be a real normed linear space and also consider that there is a 2-norm on A which makes $(A, \Vert \cdot, \cdot \Vert )$ a 2-Banach space.

Theorem 3.7

Take $\varepsilon , \theta , s \geqslant 0$ and $r \neq 1$. Let $g : A \to A$ be a function satisfying the inequality

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \textstyle\begin{cases} \varepsilon ( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} ) \Vert w \Vert ^{r}, & r>1; \\ \theta +\varepsilon ( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} ) \Vert w \Vert ^{r}, & r< 1, \end{cases} $$

(3.6)

for all $x,y , z, w \in A$. Then g satisfies functional equation (1.1); that is, $D_{g} (x, y, z)=0$ for all $x,y,z\in A$.

Proof

Two cases arise according to whether $r>1$ or $r<1$. First, take the case $r>1$. Replacing w by $\frac{w}{n}$ in (3.6), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\varepsilon }{n^{r-1}} \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert ^{r},\quad x,y , z, w \in A, n\geqslant 1.$$

So, by taking the limit as $n\to \infty$, we obtain $\Vert D_{g} (x, y, z), w \Vert =0$ for all $x,y , z, w \in A$. Then $D_{g} (x, y, z)=0$ for all $x,y , z \in A$ by Lemma 1.3.

Now consider the case $r<1$. Replacing w by nw in (3.6), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\theta }{n}+\varepsilon n^{r-1} \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert ^{r},\quad x,y , z, w \in A, n\geqslant 1.$$

It should be noted that if $r<0$, then the recent inequality satisfies for $w\neq 0$. Taking the limit as $n\to \infty$, we obtain the result. □

Corollary 3.8

Let $\varepsilon \geqslant 0$ and $g : A \to A$ be a function satisfying the inequality

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \varepsilon $$

for all $x,y , z, w \in A$. Then g is quadratic and satisfies functional equation (1.1).

Remark 3.9

It should be noted that the hyperstability Theorem 3.7 does not hold in normed spaces (see [17, Corollaries 2.3, 2.5]).

In the case of $r=1$, we have the following stability result.

Theorem 3.10

Let $\theta , \varepsilon , s \geqslant 0$ with $s\neq 2$. If $g : A \to A$ is a function satisfying the inequality

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \theta +\varepsilon \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert $$

(3.7)

for all $x,y , z, w \in A$, then there exists a unique quadratic function $Q : A \to A$ satisfying functional equation (1.1) such that

$$ \bigl\Vert Q(x) - g(x), w \bigr\Vert \leqslant \frac{2^{s} \varepsilon }{ \vert 4 - 2^{s} \vert } \Vert x \Vert ^{s} \Vert w \Vert . $$

(3.8)

Proof

Replacing w by nw in (3.7), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\theta }{n}+\varepsilon \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert ,\quad x,y , z, w \in A, n \geqslant 1. $$

Hence, by taking the limit as $n\to \infty$, we obtain

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \varepsilon \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert . $$

(3.9)

Letting $x=y=z=0$ in (3.9), we obtain $\Vert 2g(0),w \Vert =0$ for all $w\in A$. Then $g(0)=0$ by Lemma 1.3. We assume that $s<2$. Setting $y = z=0$ and replacing x by 2x in (3.9), we have

$$ \bigl\Vert g(2x)-4g(x),w \bigr\Vert \leqslant 2^{s}\varepsilon \Vert x \Vert ^{s} \Vert w \Vert , \quad x,w\in A. $$

Therefore

$$ \biggl\Vert \frac{1}{4^{n}}g \bigl(2^{n}x \bigr) - \frac{1}{4^{m}}g \bigl(2^{m} x \bigr), w \biggr\Vert \leqslant \varepsilon \Vert x \Vert ^{s} \Vert w \Vert \sum _{k=m}^{n-1} \biggl(\frac{2^{s}}{4} \biggr)^{k+1} $$

(3.10)

for all $x, w \in A$ and all integers $n\geqslant m\geqslant 0$. Hence, $\{ \frac{1}{4^{n}}g(2^{n} x) \}$ is a Cauchy sequence in A for all $x \in A$. Now, we will define a function $Q : A \to A$ by

$$ Q(x) = \lim_{n \to \infty } \frac{1}{4^{n}}f \bigl(2^{n} x \bigr),\quad x\in A.$$

Letting $m=0$ and taking the limit in (3.10) as $n\to \infty$, we obtain (3.8).

For the case $s>2$, the argument is similar. The uniqueness of Q is clearly obtained from (3.8). □

Remark 3.11

In the case $s>2$ and $\theta >0$, the stability Theorem 3.10 is not valid in Banach spaces.

4 Stability and hyperstability of (1.1) for functions $g:(A, \Vert \cdot, \cdot \Vert )\rightarrow (A, \Vert \cdot, \cdot \Vert )$

In this section, we study similar problems which we have discussed in the last section for functions $g: (A, \Vert \cdot ,\cdot \Vert ) \to (A, \Vert \cdot, \cdot \Vert )$, where $(A, \Vert \cdot, \cdot \Vert )$ is a 2-Banach space.

First, we will establish the Hyers–Ulam stability and generalized Hyers–Ulam stability of a Jensen’s quadratic functional equation (1.1), which is controlled by the sums of powers of norms on 2-Banach spaces.

Theorem 4.1

Let $g:\mathcal{A}\to \mathcal{A}$. Assume that $\varepsilon , \theta \geqslant 0$ and $\varphi :\mathcal{A}\to \mathcal{A}$ is a surjective function.

$(i)$:

If $s\in (0,2)$ and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \theta +\varepsilon \bigl[ \Vert x,w \Vert ^{s}+ \Vert y,w \Vert ^{s}+ \Vert z,w \Vert ^{s} \bigr], \quad x,y,z, w \in \mathcal{A}, $$

(4.1)

then there exists a unique quadratic function $Q : A \to A$ satisfying functional equation (1.1) such that

$$ \bigl\Vert g(x)-Q(x), \varphi (w) \bigr\Vert \leqslant \frac{\theta }{3}+ \frac{2^{s}\varepsilon }{4-2^{s}} \Vert x,w \Vert ^{s}, \quad x, w \in \mathcal{A}. $$

(4.2)

$(\mathit{ii})$:

If $s\in (2,+\infty )$ and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \varepsilon \bigl[ \Vert x,w \Vert ^{s}+ \Vert y,w \Vert ^{s}+ \Vert z,w \Vert ^{s} \bigr],\quad x,y,z, w \in \mathcal{A}, $$

then there exists a unique quadratic function $Q : A \to A$ satisfying functional equation (1.1) such that

$$ \bigl\Vert g(x)-Q(x), \varphi (w) \bigr\Vert \leqslant \frac{2^{s}\varepsilon }{2^{s}-4} \Vert x,w \Vert ^{s},\quad x, w \in \mathcal{A}. $$

Proof

$(i)$ Letting $x=y=z=0$ in (4.1), we obtain $\Vert 2g(0),\varphi (w) \Vert \leqslant \varepsilon$ for all $w\in A$. Since φ is surjective, for each $a\in \mathcal{A}$ we have $na\in \varphi (\mathcal{A})$ for all $n\in \mathbb{N}$. Then $\Vert 2g(0),na \Vert \leqslant \varepsilon$ for all $n\in \mathbb{N}$. Letting now $n\to \infty$ and applying Lemma 1.3, we infer that $g(0)=0$. Setting $y = z=0$ and replacing x by 2x in (4.1), we have

$$ \bigl\Vert g(2x)-4g(x),\varphi (w) \bigr\Vert \leqslant \varepsilon + 2^{s}\theta \Vert x, w \Vert ^{s},\quad x,w\in A. $$

Therefore

$$ \biggl\Vert \frac{1}{4^{n}}g \bigl(2^{n}x \bigr) - \frac{1}{4^{m}}g \bigl(2^{m} x \bigr), \varphi (w) \biggr\Vert \leqslant \sum_{k=m}^{n-1} \frac{\varepsilon }{4^{k+1}}+ \theta \Vert x, w \Vert ^{s} \sum _{k=m}^{n-1} \biggl(\frac{2^{s}}{4} \biggr)^{k+1} $$

(4.3)

for all $x, w \in A$ and all integers $n\geqslant m\geqslant 0$. Thus, $\{ \frac{1}{4^{n}}g(2^{n} x) \}$ is a Cauchy sequence in A for all $x \in A$. Now, we will define a function $Q : A \to A$ by

$$ Q(x) = \lim_{n \to \infty } \frac{1}{4^{n}}f \bigl(2^{n} x \bigr),\quad x\in A.$$

Letting $m=0$ and taking the limit in (4.3) as $n\to \infty$, we obtain (4.2). The uniqueness of Q is clearly obtained from (4.2).

For the case $s>2$, the argument is similar. □

Corollary 4.2

Let $g:\mathcal{A}\to \mathcal{A}$ satisfy

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \theta + \varepsilon \bigl[ \Vert x,w \Vert + \Vert y,w \Vert + \Vert z,w \Vert \bigr],\quad x,y,z, w \in \mathcal{A}.$$

Then there exists a unique quadratic function $Q : A \to A$ satisfying functional equation (1.1) such that

$$ \bigl\Vert g(x)-Q(x), w \bigr\Vert \leqslant \varepsilon \Vert x,w \Vert ,\quad x, w \in \mathcal{A}. $$

For the case $s\neq 1$, we have the following result.

Theorem 4.3

Take $\varepsilon , \theta \geqslant 0$ and $s \neq 1$. Let $g : A \to A$ be a function satisfying the inequality

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \textstyle\begin{cases} \varepsilon ( \Vert x,w \Vert ^{s} + \Vert y, w \Vert ^{s} + \Vert z, w \Vert ^{s} ), & s>1; \\ \theta +\varepsilon ( \Vert x, w \Vert ^{s} + \Vert y, w \Vert ^{s} + \Vert z, w \Vert ^{s} ), & s< 1, \end{cases} $$

(4.4)

for all $x,y , z, w \in A$ (with $\Vert x,w \Vert \Vert y,w \Vert \Vert z,w \Vert \neq 0$ when $s<0$). Then $D_{g} (x, y, z)=0$ for all $x,y,z\in A$.

Proof

First we consider the case $s>1$. Replacing w by $\frac{w}{n}$ in (4.4), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\varepsilon }{n^{s-1}} \bigl( \Vert x, w \Vert ^{s} + \Vert y, w \Vert ^{s} + \Vert z, w \Vert ^{s} \bigr),\quad x,y , z, w \in A, n \geqslant 1.$$

Taking the limit as $n\to \infty$, we obtain $\Vert D_{g} (x, y, z), w \Vert =0$ for all $x,y , z, w \in A$. Then $D_{g} (x, y, z)=0$ for all $x,y , z \in A$ by Lemma 1.3.

Now we consider the case $s<1$. Replacing w by nw in (4.4), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\theta }{n}+\varepsilon n^{s-1} \bigl( \Vert x, w \Vert ^{s} + \Vert y, w \Vert ^{s} + \Vert z, w \Vert ^{s} \bigr),\quad x,y , z, w \in A, n\geqslant 1.$$

It should be noted that if $s<0$, then the recent inequality is satisfied for all $x,y , z, w \in A$ with $\Vert x,w \Vert \Vert y,w \Vert \Vert z,w \Vert \neq 0$. Taking the limit as $n\to \infty$, we obtain the result. □

5 Conclusion

We have proved the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form

$$ g \biggl( \frac{x+y}{2} + z \biggr) + g \biggl( \frac{x+y}{2} - z \biggr) + g \biggl( \frac{x-y}{2} + z \biggr) + g \biggl( \frac{x-y}{2} - z \biggr) = g(x) + g(y) + 4 g(z) $$

in 2-Banach spaces by using the Hyers direct method.